Heat a large skillet over medium heat, and drizzle the olive oil into the hot skillet. Cook and stir 4 garlic cloves just until fragrant, about 1 minute, smashing them into the oil with a spoon as you stir. Sprinkle the chicken breasts with salt and pepper, and brown them in the hot oil until the surface is golden and the meat is no longer pink inside, 5 to 8 minutes per side. Remove the chicken breasts to a platter and keep warm.

Heat 1 tablespoon of olive oil in the skillet over medium heat, and scrape up any browned bits of flavor from the bottom of the skillet. Stir in 2 more cloves of garlic, the green, red, and yellow bell peppers, red onion, and asparagus, and cook and stir until the vegetables are brightly colored and softened, about 10 minutes. Toss the spinach leaves with the hot vegetables to just wilt the spinach. Divide the cooked vegetables between 4 serving plates, and top each plate with a chicken breast.

Stir the balsamic vinegar and brown sugar into the skillet over medium-low heat, stirring until the vinegar comes to a full boil and the sugar has dissolved. Cook the mixture until reduced and thickened, 3 to 5 minutes; spoon the balsamic sauce over each chicken breast to serve.