Gather the ingredients.

Trim larger pieces of beef from 5 to 6 pounds beef bones and cut them into 1-inch pieces. Position a rack in the center of the oven and heat to 400 F.

Put the bones and beef pieces in an extra-large roasting pan. Roast, turning a few times so the beef browns evenly, about 40 minutes.

Add 2 medium onions (peeled and quartered) , 2 large carrots (peeled and cut into 2-inch pieces) , and 2 stalks celery with leaves (cut into 2-inch pieces) to the pan. Stir in 2 tablespoons olive oil into the pan being sure to coat mixture.

Roast for 1 hour to 1 ½ hours more, turning a few times so the mixture roasts evenly.

Transfer the beef and vegetables to a large stockpot and set aside.

Pour off any excess grease from the roasting pan and place over medium heat. Add 1 tablespoons tomato paste and cook, stirring continuously, until the mixture darkens, about 2 minutes.

Add 2 cups of the water and bring to a boil over high heat. Reduce the the heat to low and simmer, scraping the brown bits from the bottom and sides of the pan.

Add the tomato paste mixture to the stockpot, along with the remaining 3 quarts of water . If the liquid doesn't quite cover the bones, add a little more water.

Add 1 large bay leaf , 3 to 4 sprigs fresh parsley , ¼ teaspoon dried thyme (or 1 to 2 sprigs fresh thyme) , and ½ teaspoon black peppercorns .

Bring to a boil over medium-high heat. Skim off any foamy scum from the top.

Reduce the heat to a simmer, partially cover, and simmer for 4 to 6 hours, or until the flavors become rich and concentrated.

Strain through a cheesecloth-lined strainer into a large bowl. Discard the solids. Let the liquid come to room-temperature before covering and transferring to the refrigerator.

When the fats solidify, skim from the surface and discard.

Ladle into 1-, 2-, or 4-cup freezer containers or jars, leaving about an inch of headspace. Refrigerate and use within four days or freeze for up to three months.