Season venison with salt and black pepper.
Heat olive oil in a large pot or Dutch oven over medium-high heat. Once the oil starts to lightly smoke, sear venison until browned. Remove and set aside.
Lower the heat to medium. Add onion and celery. Cook until tender, about 5 minutes. Add the garlic and cook for 2 minutes. Add the tomato paste and cook while stirring often for 1 minute.
Pour in about 6 cups of water or stock to cover. Add bay leaf, rosemary, and thyme. Bring to a gentle simmer.
Simmer, covered, 2 to 3 hours or until the venison is tender. Strain and reserve the broth and meat. You should have about 4 cups of liquid remaining. Freeze any extra for later use. Discard the vegetables, bay leaf, and herbs.
Add the venison to a bowl and shred it finely. Set aside.