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The phrase like a cock at a gooseberry, also like a cock at a grossart, and variants, mean: very quickly and without hesitation; eagerly; ‘like a shot’.

This phrase originated in Scotland and northern England.

In fact, grossart, groset, etc., are Scottish and northern-English forms, borrowed from the French noun groseille, designating a gooseberry.

A variant of the phrase occurred, for example, in the following from The Works of Robert Burns; With his Life (Boston (Massachusetts): Hilliard, Gray, and Company, 1835), by the Scottish poet Robert Burns (1759-1796) and by the Scottish poet and biographer Allan Cunningham (1784-1842) [Vol. 4, Glossary, page 429]:

Groset. A gooseberry.
“He lap at me like a cock at a grozet.” Scots Saying.
Grumph. A grunt, to grunt.
“What can ye get of a sow but a grumph.” Scots Prov.

These are, in chronological order, the earliest occurrences of the phrase like a cock at a gooseberry, also like a cock at a grossart, and variants, that I have found:

1-: From The Heroine of Love, A Musical Piece of Three Acts (York: Printed by W. Blanchard and Co., 1778), by James Robertson * [Act 1, page 10]:

What a fuss does she make about running away with a handsome fellow, a thing that half her sex would jump at, like a cock at a gooseberry, as the saying is.

* A comic actor, playwright and poet born in Dublin, James Robertson (1713-1795) was associated for two decades with the York Theatre, in northeastern England.

2-: From The Fortunes of Nigel, A Romance (Philadelphia: H. C. Carey and I. Lea, 1822), by the Scottish poet and novelist Walter Scott (1771-1832) [Vol. 1, chapter 14, page 242]:

“The tale of coin is complete,” said Richie, with the most imperturbable gravity; “and for the weight, though they are sae scrupulous in this town as make mouths at a piece that is a wee bit light, or that has been cracked within the ring, my sooth, they will jump at them in Edinburgh like a cock at a grossart. Gold pieces are not so plenty there, the mair the pity!”

3-: From Goslington Shadow: A Romance of the Nineteenth Century (New York: Published by Collins and Hannay, Collins and Co., [&c.], 1825), a novel set in Scotland, by Mungo Coultershoggle [Vol. 1, chapter 7, page 73]:

They went to Goslington’s lodgings. To their sorrow, he was sick a bed from a violent headach [sic]. He had for several nights sat up very late, and by extreme exertion and want of rest, had produced this severe attack. During the time the visiters were at breakfast, the Domine’s ingenuity suggested to him, that Goslington might be personified by his sister, who for once in her life had the pleasure to hear from her quondam teacher, that she was even superior to her brother as a Latin scholar.
“My princeps eruditionis,” said the Domine to Peggy, “I waud loup like a cock at a groset, to see my Minerva of the Kype, carry off the prize at the Blackstone examination as your brother is too unwell to attend.”

4-: From The Man-of-War’s Man, by the Scottish author John Howell (1788-1863), published in Blackwood’s Edinburgh Magazine (Edinburgh, Midlothian, Scotland) of July 1825 [chapter 14, page 43, column 1]:

“I hae three lads o’ my ain,
Though they’re no that quite comely to see,
They’re better than them that has nane,
And wad loup like a cock at a grossart at me.
The tane he is lame o’ a leg,
The tither is blind o’ an e’e,
And the third has a tea-kettle back—
But they’re a’ a-courting o’ me.
Then Alloa lads for me!” &c. &c.