Heat oil in a large Dutch oven over medium-high. Add onion; sauté 3 minutes. Add 1/4 teaspoon salt, crushed red pepper, and garlic; cook 1 minute. Stir in tomato paste; cook 1 minute. Add stock, torn basil leaves, and tomatoes; bring to a simmer. Cook 8 minutes. Stir in remaining 1/4 teaspoon salt, sugar, and freshly ground black pepper.
Place tomato mixture in a blender. Remove center piece from blender lid (to allow steam to escape); secure lid on blender. Place a clean towel over opening (to avoid splatters); blend until smooth. Place 1/4 cup soup and yogurt in a small bowl; stir until smooth. Stir yogurt mixture into remaining soup. Divide soup among 4 bowls; top evenly with chopped basil.