Preheat oven to 350°F. Warm oil in a large skillet over medium heat. Cook onion, stirring, about 3 minutes. Add garlic; cook 2 minutes more, until onions are soft. Crumble in turkey and stir until no longer pink, about 5 minutes. Add tomatoes; season with salt and pepper. Cook, stirring, until most of liquid is absorbed, about 10 minutes. Transfer to a large bowl.
Separate biscuits and roll each out into a 6-inch circle on a floured counter.
Use a pastry brush or your fingers to wet rim of each biscuit with a bit of water. Spoon 1/3 cup turkey mixture on half of a biscuit circle and top with 1 Tbsp. cheese. Fold dough over filling, pressing wet edges to seal tightly. Crimp edges with a fork. Slice a small vent in top of each.
Place on a large baking sheet; bake until golden brown, 20 to 25 minutes.