Pat chicken dry and pound to an even 1/2-inch thickness.
Season both sides with salt and pepper, then dust lightly with flour.
Heat olive oil and 1 tbsp butter in a large skillet over medium-high.
Sear chicken 4–5 minutes per side until golden. Transfer to a plate.
Reduce heat to medium. Melt remaining butter and sauté garlic for 30 seconds.
Pour in broth and scrape up browned bits. Simmer for 2 minutes.
Stir in cream, Parmesan, Italian seasoning, and red pepper flakes. Simmer to thicken slightly.
Nestle chicken back in. Spoon sauce over and simmer until 165°F, 5–7 minutes.
Finish with lemon juice and parsley. Taste and adjust salt and pepper.