To prepare the pigeons for cooking, remove the wishbone, then spatchcock the birds (cut down both sides of the backbone and remove it, then flatten the birds). Trim off excess ribcage. Set aside.
For the garlic confit, put the garlic cloves in a saucepan and cover with the melted butter. Add the thyme, bay leaf, and salt and pepper to taste. Cook gently for 15-20 minutes to soften the g