Pat chicken dry using paper towels. Sprinkle evenly with black pepper and 1 tablespoon salt. Place chicken, skin side down, in a 12-inch skillet over medium-high. Cook, undisturbed, until fat is rendered and chicken skin is golden brown and crisp, 12 to 14 minutes. Flip chicken. Add oil to skillet; continue cooking chicken, undisturbed, until a thermometer inserted into thickest portion of chicken registers 170°F, 8 to 10 minutes. Transfer chicken, skin side up, to a plate.

Add bell peppers, onion, and garlic to skillet with chicken drippings; cook over medium-high, stirring occasionally, until softened and lightly browned, about 8 minutes. Add tomatoes; cook, stirring occasionally, until softened and liquid has evaporated, about 8 minutes. Remove from heat; stir in sherry vinegar, brown sugar, piment d’Espelette, and remaining ¾ teaspoon salt. Transfer mixture to a blender; secure lid, and remove center piece to allow steam to escape. Place a clean towel over opening, and process until smooth, about 1 minute.

Pour vegetable puree into skillet, and bring to a simmer over medium. Cover and cook, stirring occasionally, until sauce thickens slightly, 6 to 8 minutes. Return chicken thighs to skillet, skin side up. Cook, uncovered, over medium-low until warmed through, about 5 minutes.

Garnish chicken with additional black pepper and piment d’Espelette before serving.