Place the chicken breasts between two sheets of plastic wrap or in a zip-top bag. Pound them to about ½-inch thick so they cook evenly. Season both sides with salt and pepper to start building flavor right away.
Heat olive oil in a skillet over medium-high heat. Cook the chicken for 5–7 minutes per side until golden brown and fully cooked. Transfer to a plate and keep the skillet as-is—the browned bits will flavor the sauce.
Reduce the heat and melt the butter in the same skillet. Add the shallots and garlic, cooking until soft and fragrant. This forms the aromatic base of your sauce.
Pour in the half-and-half and chicken broth, stirring while scraping up the browned bits. Add the Parmesan slowly, letting it melt and thicken the sauce. You should have a creamy, garlicky sauce at this point.
Return the chicken to the skillet and spoon sauce over each piece. Let it simmer for 2–3 minutes so the chicken soaks up the flavors. Finish with fresh parsley on top for color and freshness.