Process onion, carrot, mushrooms, and garlic in food processor into 1/8-inch pieces, 10 to 12 one-second pulses, scraping down sides of bowl as needed.

Heat oil in Dutch oven over medium-high heat until simmering; add beef and tomato paste and cook, stirring frequently, until beef is well browned, 8 to 10 minutes. Add vegetable mixture and cook, stirring occasionally, until any released moisture has evaporated, about 8 minutes.

Add wine and bring to simmer, scraping bottom of pan with wooden spoon to loosen browned bits. Add beef broth, thyme, bay leaves, and peppercorns; bring to boil. Reduce heat and gently boil, occasionally scraping bottom and sides of pot, skimming fat from surface, until reduced to 2 cups, 20 to 25 minutes.

Strain mixture through fine-mesh strainer set over small saucepan, pressing on solids with rubber spatula to extract as much liquid as possible (you should have about 1 cup stock). Sprinkle gelatin over stock and stir to dissolve. Place saucepan over medium-high heat and bring stock to boil. Gently boil, stirring occasionally, until reduced to 3/4 to 1/2 cup, 5 to 7 minutes. Remove from heat and cover to keep warm.