Heat a large deep skillet or pot over medium heat. Add the butter. When melted, add the onions, celery, and carrots. Sweat until the onions are translucent, about 10 minutes. Do not brown.
Add the mushrooms and garlic. Cook until the mushrooms just begin to soften. Around 5 minutes.
Add the venison, rosemary, thyme, caraway, and tomato paste. Season with salt and pepper. Stir to combine and cook while stirring for 5 minutes.
Sprinkle in the flour evenly and stir so there are no clumps. Increase the heat to medium-high and slowly add the dark beer while stirring.
Pour in enough stock to just cover the mixture. Bring to a gentle simmer, cover, and cook until the venison is fork-tender, 2 to 3 hours. Stir and check liquid levels every 30 minutes.The mixture should be thick when cooking is complete. If not, remove the lid and simmer until the mixture is thick enough to coat the back of a spoon. Season to taste with salt and pepper. Let cool slightly and remove the bay leaf.
While the mixture is cooling, preheat the oven to 350°F.