Warm the olive oil and butter in a deep skillet over medium heat. Add the minced garlic, sautéing briefly for about 30 seconds until aromatic.
Incorporate the diced tomatoes, tomato paste, chicken broth (or white wine), and Italian seasoning into the skillet. Stir to combine.
Add the uncooked gnocchi directly to the pan. Cook for 5-7 minutes, stirring frequently to prevent sticking, until the gnocchi is tender and the sauce thickens from the released starch.
Taste the gnocchi to ensure doneness. Stir in the mozzarella, parmesan, and fresh basil, mixing until the cheeses melt smoothly into the sauce. Adjust seasoning with salt and pepper as needed. Serve immediately.