Preheat the oven to 400°F and position a rack in the middle.
Slice each chicken breast in half lengthwise to make 4 thinner cutlets.
Season both sides of the cutlets with garlic powder, chili powder, salt, and pepper, then lightly coat each piece in flour, shaking off excess.
Heat the butter and olive oil in an oven-safe skillet over medium-high heat until shimmering.
Sear the chicken 3–4 minutes per side, until lightly golden; remove the chicken and set aside.
Pour the chicken broth into the skillet and scrape up any browned bits, then stir in the salsa until combined.
Return the chicken to the skillet, spoon some salsa over each piece, and sprinkle the shredded cheese evenly on top.
Transfer the skillet to the oven and bake 5–7 minutes, until the chicken is cooked through and the cheese is melted; optionally broil 1–2 minutes to brown the cheese, watching carefully.
Garnish with chopped cilantro if desired and adjust seasoning with extra salt and pepper before serving.