Place chicken breasts between plastic wrap and pound thicker parts to an even 1/2-inch thickness.
Season the flour in a shallow dish with 1/2 teaspoon salt and 1/2 teaspoon pepper, then dredge both sides of each chicken breast in the seasoned flour, shaking off excess.
Heat 2 tablespoons olive oil in a 12-inch skillet over medium-high heat until shimmering.
Sear the chicken in the hot skillet until cooked through and an instant-read thermometer reads 165°F (about 6 minutes per side), then transfer to a plate and tent with foil.
Reduce heat to medium, add the minced garlic to the skillet, and sauté until just lightly golden, about 30 seconds.
Pour in 1 cup chicken broth, add the lemon juice and 1/2 teaspoon dried basil, and simmer until the liquid reduces by about half, about 2 minutes.
Whisk 1 tablespoon reserved chicken broth with 2 1/2 teaspoons cornstarch, then slowly stir it into the simmering broth and cook, stirring constantly, until slightly thickened, about 30 seconds.
Stir in 1/2 cup heavy cream, reduce heat to low, then add the Pecorino Romano and cook just until the cheese melts and the sauce is smooth.
Return the chicken to the skillet and spoon sauce over the pieces to coat and warm through, then remove from heat.
Serve the chicken warm, garnished with chopped parsley.
