Place each chicken breast between plastic wrap and pound the thick parts to an even 1/2-inch thickness.
Heat 2 Tbsp olive oil in a 12-inch skillet over medium-high heat until shimmering.
In a shallow dish whisk together 1/4 cup all-purpose flour with about 1/2 tsp salt and 1/2 tsp black pepper.
Dredge both sides of each chicken breast in the flour mixture, shaking off excess, then add to the hot skillet.
Cook the chicken until cooked through and an instant-read thermometer reads 165°F in the center, about 6 minutes per side; transfer to a plate and tent with foil.
Reduce heat to medium, add the minced garlic to the skillet and sauté until just light golden, about 30 seconds.
Add 1 cup chicken broth, 2 Tbsp lemon juice, and 1/2 tsp dried basil to the skillet and bring to a simmer.
Simmer the broth mixture until reduced by about half, about 2 minutes.
Whisk 1 Tbsp chicken broth with 2 1/2 tsp cornstarch in a small bowl, then pour into the simmering sauce and stir constantly until slightly thickened, about 30 seconds.
Stir in 1/2 cup heavy cream, reduce heat to low, add the shredded Pecorino Romano, and cook just until the cheese melts into the sauce.
Return the chicken to the skillet and spoon the sauce over the pieces to warm through, then sprinkle with chopped parsley and serve.