Position a rack in the center of the oven and heat the oven to 450°F.
Put the bones on a large rimmed baking sheet and roast until beginning to brown, about 20 minutes. Add the carrots and onions and continue roasting until the bones and vegetables are very brown, 30 to 45 minutes more.
With a slotted spoon or tongs, transfer the roasted bones and vegetables to a stockpot, leaving any rendered fat in the pan. Add the bouquet garni, peppercorns, tomato paste, and 5 to 7 quarts cold water (enough to cover the bones and vegetables by a couple of inches) to the pot. Bring to a boil slowly over medium heat, reduce the heat to medium low or low, and simmer, uncovered, skimming the surface occasionally with a slotted spoon until the broth is flavorful and reduced enough to just barely cover the bones and vegetables, 4 to 5 hours.
Strain the broth into a large bowl, cover, and chill. Skim off any fat before using.
The broth can be refrigerated for up to 3 days or frozen for up to 2 months.