Place the flour, ¾ teaspoon salt, and the black pepper in a ziplock bag and shake to combine.
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Add the chicken to the bag; seal the bag tightly and shake to coat the chicken evenly. Set aside.
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Heat the oil in a large nonstick skillet over medium-high heat. Place the flour-dusted chicken in the pan, shaking off any excess first.
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Cook, flipping once, until the chicken is lightly golden and just cooked through, about 3 minutes total. Transfer the chicken to a plate and set aside.
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Reduce the heat to medium and add the onions to the skillet. Cook, stirring frequently, until softened and translucent, 2 to 3 minutes. Add the garlic and cook, stirring constantly, for 30 seconds more; do not brown.
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Add the cream, broth, sun-dried tomatoes, tomato paste, oregano, red pepper flakes, sugar, and ¼ teaspoon salt; use a wooden spoon to scrape any brown bits from the pan into the liquid.
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Bring the liquid to a boil, then reduce the heat to medium and gently boil, uncovered, until the sauce is thickened, about 5 minutes.
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Stir in the Parmigiano Reggiano (or Pecorino Romano) and basil.
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Add the chicken back to the pan, along with any juices that accumulated on the plate; reduce the heat to low and simmer until the chicken is warmed through and the sauce thickens a bit more, 1 to 2 minutes.
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Sprinkle with more fresh basil, if desired, and serve.