Pat venison pieces dry with paper towel and season with salt and pepper.
Heat 1 tablespoon oil in a large stockpot or Dutch oven over medium-high heat until just smoking. Add half the meat in a single layer and cook, without moving, until browned on one side, about 5 minutes.
Flip each piece of venison and continue cooking until browned on the other side. Transfer to a bowl. Heat another tablespoon of oil and repeat with remaining venison until browned. Transfer to bowl.
Heat last tablespoon of oil until shimmering. Add onion and celery and cook until softened, about 5 minutes. Stir in garlic until fragrant, about 30 seconds.
Stir in flour and cook until lightly browned, about 2 minutes. Add tomato paste, wine, and chicken broth, thyme, bay leaves, and browned venison, scraping up any browned bits on the bottom of the pan.
Bring to a boil. Reduce heat, cover, and simmer for 1 hour.
Stir in potatoes and carrots. Return to a boil, reduce heat, cover, and simmer 1 hour longer, until venison is tender.
Remove bay leaves and any thyme stems. Stir in peas and cover for 5 minutes. Season to taste with salt and pepper