Melt bacon drippings in a large Dutch oven over medium-high heat. Add venison; sear, in two batches, until nicely browned. Transfer venison to a plate.

Add onions, shiitakes, and garlic; cook until softened, 1 to 2 minutes. Stir in wine, cognac, and bouillon cube, simmer 30 seconds to remove alcohol flavor and dissolve bouillon.

Add venison, bay leaves, rosemary, thyme, and black pepper; bring to a boil, then reduce heat to low. Gently simmer until venison is tender, about 2 hours, adding water as needed.

Add baby carrots during the last 30 minutes of cooking. Dissolve cornstarch in 2 tablespoons water when venison is tender; stir into sauce to thicken.