Season both sides of the chicken breasts with a light sprinkle of salt, pepper, and about 1 teaspoon of dried oregano.
Warm 1 tablespoon of the olive oil in a large skillet over medium-high heat. When hot, place chicken in the skillet and cook for 2-3 minutes on each side, until it is cooked through and golden-brown on the outside. Remove chicken to a plate and set aside.
While the chicken cooks, add the roasted peppers, remaining 1 teaspoon oregano and 1 tablespoon olive oil, garlic, 1/2 teaspoon salt, and 8-10 cranks of black pepper to a blender or food processor. Blitz until very smooth.
After removing the chicken, add an extra drizzle of oil if the pan looks very dry. Add shallots and cook for 2-3 minutes, until beginning to soften. Pour in the red pepper mixture and scrape the bottom of the pan to loosen any browned bits. Simmer for 2-3 minutes to thicken slightly, then stir in the cream and cook 1 minute more, just to warm through.
Return chicken to the pan and spoon sauce on top. Garnish with Parmesan and basil, plus extra chopped red peppers if desired. Enjoy!
Notes
This makes a good amount of sauce for the chicken and soaking a little into some bread or mixing with veggies served on the side. If you plan to serve with pasta and want to use the sauce to coat the pasta, as well, consider upping the amounts to 1.5 cups red peppers and 1/2 cup heavy cream, and measure the other ingredients as heaping spoons.