Preheat the oven to 325°F.
Pat the venison dry and season with salt and pepper.
Heat olive oil in a large Dutch oven over medium-high heat. Working in batches, sear the venison on all sides until browned. Transfer to a plate.
Reduce heat to medium. Add the chopped onion and sauté for 5 minutes, until softened.
Stir in the garlic and cook just until fragrant.
Sprinkle in the flour and stir to coat the onions.
Add the tomato paste and cook for 1 minute to deepen the flavor.
Pour in the beef broth and stir well to combine.
Return the venison to the pot. Add the carrots, celery, potatoes, thyme, and bay leaves. Stir to combine.
Bring to a gentle simmer, then cover and transfer to the oven.
Braise for 2 to 2½ hours, or until the venison is fork-tender and the vegetables are soft.
Remove the bay leaves, taste, and adjust seasoning if needed.