Literally translated as, being like a rooster in dough, être comme un coq en pâte means feeling cosy and pampered, being in a state of absolute contentment, with one’s every need catered to. It's close enough to the English idiom, “to be in clover” except that the latter refers primarily to financial comfort, whereas the French expression implies a more general sense of physical and spiritual well-being.
Coq en pâte is an olden, luxurious French dish in which a poularde (fatted chicken) is stuffed, trussed, wrapped entirely in a short crust, and then baked until golden. It is traditionally served with sauce Périgueux on the side, a sauce flavored with port and truffles.
Note that le coq (rooster) and la coque (shell) are 2 homophones not be confused. If someone asks if you want to have un œuf à la coque for le petit déjeuner (breakfast), they are not asking you to have a “rooster egg”, but simply a soft-boiled egg.
The idea behind this 17th century idiom is that the bird, while it is being prepared for this recipe — handled with loving care, adorned with prized ingredients, and bundled up snugly in a buttery blanket — must feel really, really good.
One might argue that its bliss comes a tad late for the bird to actually enjoy, but this idiom is perhaps a good illustration of how the French popular wisdom considers it an honorable thing for an animal to finish its life as a delicious dish on a plate.
If you’re curious, or maybe in need of inspiration for your Holiday menu, here is the recipe for Coq en pâte.
Depuis qu'il est à la retraite, il vit comme un coq en pâte.
Since he retired, he has been living a good and easy life.
Quand il est chez ses grand-parents, notre fils est comme un coq en pâte.
When he stays with his grand parents', our son is treated like a king.