In a large skillet over medium heat, add olive oil. Add onion and sauté, stirring, until soft, about 3 minutes. Add minced garlic and sauté with onions another 1 minutes.
Add ground venison and cook, stirring and breaking up meat with wooden spoon until no longer pink, about 5 minutes. Add salt, pepper, and paprika and mix well.
Stir in ketchup, mustard, Worcestershire, brown sugar, and chili powder. Simmer until thick, about 4 minutes. Transfer mixture to a plate and wipe skillet clean.
Assemble sandwiches
Butter one side of each slice of Texas Toast and pile on ¼ cup of shredded cheese on the opposite side. Then spoon some of the sloppy joe mixture on top of the cheese. Then top that with another ¼ cup of shredded cheddar cheese and one more slice bread, buttered-side up.
Cook over medium heat until bread is golden brown, and cheese is melted, about 3 minutes per side. Repeat with remaining ingredients to make 4 sandwiches total.