Preheat oven to 425F/215C.
Heat a shallow braiser to medium heat and add 1 Tbsp olive oil. Add bacon and fry until crisp on both sides, about 5-6 minutes total. Remove and set aside.
Season chicken thighs generously with salt and pepper on both sides. Add to the hot pan, skin side down. Cook until browned and crispy, about 4-5 minutes on the first side. Flip to the other side and cook until brown and crispy, about 2-3 more minutes. Remove chicken from the pan and set aside on a plate. It's ok if the chicken is not fully cooked; the chicken will finish cooking in the oven.
If the pan has an excessive amount of oil, remove all but 2 Tbsp. In the same pan, add anchovies. Stir and break the anchovies into smaller pieces with a wooden spoon, about 1 minute. They should melt into the oil, leaving a few pieces behind.
Add onion, garlic, capers, and dried chili flakes. Mix with a wooden spoon until onion has softened and looks translucent, about 5 minutes. Add tomatoes and break down with wooden spoon (or potato masher). Cook for 10 more minutes until thick and rich, stirring from time to time.
Add chicken, skin side up, directly on top of the tomato sauce. Transfer to the oven and bake until cooked through, about 10-12 minutes.
Remove shallow braiser from the oven and turn off heat. Add cubed mozzarella in between chicken pieces, directly into the sauce. Transfer to the oven and let the residual heat melt the mozzarella gently into gooey puddles, about 5 minutes.
Remove from the oven and garnish with basil. Chop bacon into 1-inch strips and scratter evenly over the top. Sprinkle more dried chili pepper flakes, if desired, and a few more grinds of black pepper. Eat immediately, preferably with crusty bread!