Heat olive oil in a pot over medium heat. Cook and stir garlic in hot oil until fragrant, about 2 minutes. Break ground turkey into the pot. Cover pot with lid and cook meat until browned on the outside, 2 to 3 minutes. With a slivered wooden spoon, break meat up into very small pieces; continue to cook and stir until nearly completely browned, 3 to 5 minutes.

Stir tomato, onion, green bell pepper, parsley, and red pepper flakes into the turkey mixture; season with salt and black pepper; cook until the tomato softens, 3 to 4 minutes. Add tomato past and lemon juice; stir to coat vegetables with tomato paste. Cover pot with lid, reduce heat to low, and cook until the vegetables are tender, 7 to 10 minutes. Serve on toasted hamburger buns topped with cheese.