The humorous phrase the thick plottens is a deliberate transposition of the initial consonants of plot and thickens in the plot thickens. This type of transposition is called spoonerism—as mentioned in the following from The Illustrated London News (London, England) of Saturday 20^th December 1952:

There is a rather awkward moment in “An Italian Straw Hat” when Laurence Payne, as a young bridegroom, looking desperately into the auditorium of the Old Vic, cries: “The thick plottens!” Hearing this elementary Spoonerism, graver members of the audience at the première bent their heads, and one seemed to detect a sound of low moaning. It was not, I fear, an inspired fizz of wit; but these things do happen in farces; the moans came, I fancy, merely because the farce was acted on the grave stage of the Old Vic. Otherwise, none would have cared a straw.

The phrase the plot thickens means the storyline becomes more complex or convoluted, and is first recorded in The Rehearsal, As it was Acted at the Theatre-Royal (London: Printed for Thomas Dring, at the White-Lyon, next Chancery-lane end in Fleet-street. 1672), by George Villiers (1628-1687), 2^nd Duke of Buckingham:

Iohns. But, Mr. Bayes, is not that some disparagement to a Prince, to pass for a Fishermans Son? Have a care of that, I pray.
Bayes. No, no, no; not at all; for ’tis but for a while: I shall fetch him off again, presently, you shall see.
Enter Pretty-man and Thimble.
Pret. By all the Gods, I’l set the world on fire
Rather than let ’em ravish hence my Sire.
Thim. Brave Pretty-man, it is at length reveal’d,
That he is not thy Sire who thee conceal’d.
Bayes. Lo’ you now, there he’s off again.
Iohns. Admirably done i’faith.
Bayes. Ay, now the Plot thickens very much upon us.
Pret. What Oracle this darkness can evince?
Sometimes a Fishers Son, sometimes a Prince.
It is a secret, great as is the world;
In which, I, like the soul, am toss’d and hurl’d.
The blackest Ink of Fate, sure, was my Lot.

These are the earliest occurrences of phrase the thick plottens that I have found, in chronological order:

1-: From the Daily Illinois State Register (Springfield, Illinois) of Friday 14^th December 1883:

MEN WITH “CLEWS.”
The Detective Talent of Lincoln “Hauled Over the Coals.”

Special Correspondence State Register.
Lincoln, Dec. 13.—“The thick plottens.” The detective force (?) is each day augmented and increased by the arrival, from some cross-roads, of a “slick” man with a “clew.” Each train brings one or more of these “ducks.” You can tell them by their penetrating glance and sort of what-der-yer-sov air and John L. Sullivan gait. The white sombrero with the rim pulled down in front and its owner looking out from under his eyelids as though he knew a thing or two, are symptoms of detectiveania, a new disease that has lately broken out here. If a man is seen with high leg boots, pants tucked inside, a sort of Dr. Carver of Buffalo Bill cut to his coat, he is at once “marked,” he is a “fly copper” sure, and Lincoln is as full of them as a dog is of fleas. What does this all mean? It means that Carpenter, guilty or innocent, is pursued by a gang of sleuth hounds whose chief characteristic is to get money at any sacrifice, and look wise.

2, 3, 4 & 5-: From In the Wake of the News, by ‘Hek’, published in The Chicago Tribune (Chicago, Illinois):

2-: Of Thursday 5^th September 1907:

Our war correspondent at Dubuque writes: “The presence of Comiskey’s 25 h. p. armored battleship White Sox in the Dubuque harbor attaches pointed significance to the Three Lamp League crisis. When I called at eleven bells the lights were doused, the commander had turned in, and the officer of the day had gone ashore to lap up a few (in technical parlance). I have it on good authority that the White Sox is here to help the citizens protect the franchise. The thick plottens.” (More.)

3-: Of Friday 22^nd November 1907:

Mr. Corrigan was not to be seen yesterday, having retired to his country place, but he left word with his counsel to the effect that he feared nothing from Germany, now that he has formed a friendly alliance with Italy, to which, before the organization of the American Turf association, he was strongly—or strongly was—opposed. It is not likely in any event that he will willingly withdraw the left hand lead he handed Herr Dokter Frank or that he will abate one jot of his right to consider Herr Dokter Frank one Prussian mutt that doesn’t know any more about ring bones than Doc Talbot does about picking winners.
Meanwhile the thick plottens.

4-: Of Saturday 19^th September 1908:

According to dispatches received by our Mr. C. F. Spalding from various points in the east, the Wells goat is still at large. It escaped from its keepers at Bangor, Me., and was last seen swimming the St. Lawrence river with the crawl stroke. The thick plottens.

5-: Of Wednesday 3^rd February 1909:

The thick plottens in the checker scandal. Our expert along those lines, whose monicker is thinly veiled in the subscription to the appended document, evidently knows more than he is of a mind to spring at this time.

6-: From A Few Paraphrases, by Wex Jones, published in The San Francisco Examiner (San Francisco, California) of Wednesday 7^th September 1910:

The combat deepens. The thick plottens. On, ye guys who are looking for glory or a six-foot lot. Wave, Munich, all thy schoppens wave, and charge with all thy nerve. Tough, tough, for many a chap shall fall in the snow.