Preheat the oven to 350°F (175°C).
Heat the olive oil in a large skillet over medium heat, then add the diced onion and cook until softened, about 4–5 minutes.
Add the minced garlic and the sliced chicken breasts to the skillet and cook, stirring occasionally, until the chicken is browned and cooked through, about 6–8 minutes (depending on thickness).
Stir in the salt, cayenne pepper, and balsamic vinegar, cook for 1–2 more minutes to combine, then remove from heat.
Place the halved bell peppers cut-side up in a baking dish and spoon the chicken mixture into each pepper half, dividing it evenly.
Pour a few tablespoons of water into the bottom of the baking dish, cover tightly with foil, and bake for 35 minutes.
Remove the foil, top each pepper half with a half slice of provolone, return to the oven uncovered, and bake until the cheese is melted and bubbly, 5–10 minutes.