Heat 3 tablespoons of the olive oil in a large skillet over medium-high heat. Add the diced onions and peppers, and saute them for approximately 5 minutes, or until translucent and beginning to caramelize. Add the garlic and saute another minute. Stir in 1/2 teaspoon of the salt, 1/2 teaspoon of the pepper, 1/2 teaspoon of the rosemary, and 1/2 cup of the chicken broth. Bring the chicken broth to a boil, then remove the pan from the heat. Transfer the broth mixture to a medium bowl and add the breadcrumbs; stir the breadcrumbs until fully incorporated into the broth. Add the eggs and asiago cheese, and stir this well also. Spread the stuffing mixture to the sides of the bowl (to create more air surface for quick cooling) and immediately place the bowl in the refrigerator until thoroughly chilled.
Place the chicken breast skin-side down between two pieces of waxed paper or cellophane, and using the smooth side of a meat mallet or bottom of a small pan, gently pound the chicken to flatten it slightly, which will make them slightly larger in diameter and create an even thickness.
Preheat an oven to 375 degrees Fahrenheit. Remove the top piece of waxed paper or cellophane, and divide the chilled stuffing amongst the pounded chicken breasts. Fold the chicken around the stuffing and turn it skin-side up. Secure the chicken breast around the stuffing by tying a piece of twine around each breast.
Place the stuffed chicken breasts on a rack that fits properly into a roasting pan. Brush the skin of the chicken with the remaining 3 tablespoons olive oil and place them into the preheated oven. Roast the chicken breast for approximately 30-40 minutes, or until the skin is crisp and the internal temperature of the center of the stuffing reads 160 degrees Fahrenheit. Remove the chicken and the rack from the roasting pan, and set aside.
Pour off and fat or juices that may have accumulated, and add 2 tablespoons of fat back to the roasting pan (rendered chicken fat, butter, or olive oil). Place the roasting pan over a burner with medium heat and stir in the flour. Stir the fat and flour mixture for a couple of minutes, taking great care not to burn it, then add the remaining 1/2 teaspoon salt, 1/2 teaspoon pepper, and 1 cup chicken broth. Bring the broth to a simmer, scrape any caramelized pieces of the bottom of the pan, and simmer it for 2 or 3 minutes. Remove any visible pieces or lumps by straining the gravy through a fine strainer or cheesecloth. Drizzle over the chicken on a serving platter, or serve in a sauceboat on the side.