Mince garlic with ¼ teaspoon salt to form smooth paste. Transfer to large bowl. Add 2 tablespoons oil, vinegar, anchovies, thyme, lemon juice, pepper flakes, paprika, and ½ teaspoon salt. Whisk to combine.
Heat 12-inch cast-iron skillet over medium heat for 10 minutes. Toss tomatoes with 2 teaspoons oil and ½ teaspoon salt in medium bowl. Add tomatoes to skillet and cook until lightly charred on 1 side, about 4 minutes. Stir and continue to cook 2 minutes longer. (Some tomatoes will start to burst.)
Meanwhile, toss bell pepper with remaining 2 teaspoons oil and remaining ½ teaspoon salt.
Transfer tomatoes to vinegar mixture. Add bell pepper to now-empty skillet set over medium heat. Cook until spotty brown on 1 side, about 4 minutes. Stir and continue to cook 2 minutes longer. Transfer bell pepper to tomato mixture and toss gently to combine. Let sit for at least 30 minutes or up to 2 hours to allow flavors to meld.
Gently toss mozzarella and basil with tomato mixture until combined. Season with salt and pepper flakes to taste. Transfer to platter. Serve drizzled with oil.
