Heat bacon grease in an oven-proof pot or Dutch oven over medium heat. Cook and stir venison in the hot grease until browned on all sides, about 5 minutes. Transfer venison to a plate using a slotted spoon. Add onion to hot grease; cook and stir until tender, 4 to 5 minutes.
Mix garlic, tomato paste, and anchovy paste together in a small bowl; add to onions. Stir flour into onion mixture until dissolved.
Pour wine, beef broth, and chicken broth into the pot; bring to a boil while scraping the browned bits of food off of the bottom of the pot with a wooden spoon. Add thyme and bay leaves to pot and bring liquid to a simmer. Return venison to pot. Cover pot.
Simmer stew in the preheated oven for 1 hour. Add potatoes, carrots, and mushrooms to stew and simmer in the oven until potatoes are tender, about 1 hour more. Add peas and stir.
