Trim the birds, removing the heads, feet, and first joints of the wings. (If your birds come without heads and feet, ask for them from your butcher.) Remove the livers and hearts from the cavities and discard. Rinse the birds well under cool running water. Set the birds aside.
FOR THE SAUCE:
In a medium-size saucepan over medium-high heat, heat the olive oil.
Add the bird trimmings along with the salt and black pepper and cook, stirring, until well-browned, about 15 minutes. Add the onions, carrots, celery, bay leaves, and thyme and cook, stirring to loosen any brown bits from the bottom of the pan, for 5 minutes. Add the tomato paste, stir to mix, and cook for 2 minutes. Add the wine, stir, and cook for 2 minutes. Add the water, bring to a boil, reduce the heat to medium, and simmer for 1 hour. Strain through a fine-mesh sieve into another saucepan or a gravy boat. Skim off any fat that rises to the surface.
Preheat the oven to 400 degrees F.
Rub the birds with the olive oil and season with the salt and black pepper. Put the birds breast side down on a rack fitted over a baking sheet. Bake until golden brown, about 25 minutes (and the internal temperature is 150° F). Remove from the oven and let sit for 10 minutes.