Season the chicken. Pat the chicken dry with a paper towel, then season lightly on both sides with 1 tsp Kosher salt and ¼ tsp black pepper.
Brown the chicken skin. Heat ⅓ cup olive oil in a large skillet over medium-high heat. When hot, add chicken thighs, skin side down. Cook until golden brown and crispy (the chicken should very easily pull away from the pan), about 5-7 minutes.
Flip the chicken. Use tongs to carefully flip the chicken thighs over, then cook an additional 4-5 minutes, or until browned on the second side. Transfer chicken to a plate.
Flavor the sauce. To the same pan, add 6 cloves thinly sliced garlic, 4 anchovy fillets, 2 Tbsp capers, and ¼ tsp crushed red pepper flakes. Cook, stirring often, until garlic is lightly browned and anchovies have dissolved into the sauce, about 2 minutes.
Add tomatoes. Stir in 1 Tbsp tomato paste, breaking up with a spatula until dissolved, then pour in whole tomatoes. Crush with the back of a wooden spoon or spatula, breaking them up into smaller pieces. Add ½ cup water and 1 cup olives, then stir to combine.
Finish cooking chicken. Return the chicken thighs to the skillet, skin side up. Reduce the heat to a simmer, then cook until chicken registers an internal temperature of 165°F/74°C and the sauce has thickened, about 20 minutes.
Garnish and serve. Sprinkle with finely chopped parsley and fresh basil, then serve immediately.