Pat the chicken dry and season liberally with salt. Set aside.
Heat 1 Tbsp. olive oil in a Dutch oven over medium heat. Add the onions and cook, stirring frequently until just starting to soften; 2 minutes. Do not brown. Add the bell peppers and cook until just beginning to soften; another minute. Add the garlic and red-pepper flakes and season lightly with salt. Cook, stirring until the garlic is fragrant and the onions just start to brown slightly; another 2 minutes. Move the cooked veg to a plate and set aside.
Add another Tbsp. of olive oil to the Dutch oven. Place the chicken skin-side down and sear without moving until the skin is golden brown; 6 to 8 minutes. Turn and cook the other side until lightly browned, about 3 minutes. Pour out all but 1 Tbsp. fat from the pan.
Add 1/2 cup chicken stock to the Dutch oven and stir scraping up any loose bits. Stir in the sherry vinegar and honey. Cover, turn down the heat to medium-low and cook until the chicken is cooked through; about 10 minutes. Uncover, raise the heat to medium-high and cook until the sauce is thickend; about 2 to 3 minutes more.
Add the cherry tomatoes and cook until they begin to soften and burst; about 3 to 4 minutes. Return the onion-pepper mixture to the Dutch oven, stir to coat in the pan sauce and cook uncovered, until warmed through. Add 3 Tbsp. basil and stir to incorporate. Taste and add more salt if needed. Remove from the heat. Drizzle more olive oil over the top and garnish with more fresh chopped basil.