Preheat the broiler with an oven rack set about 4 inches from the heat source.
Melt the butter in a small saucepan over medium heat, swirling the pan occasionally, until light brown specks start to form and the butter starts to smell nutty, about 5 minutes. Pour immediately into a large mixing bowl, being sure to scrape the brown bits from the bottom of the pan.
Spread the peppers in a single layer on a baking sheet. Broil, shaking the baking sheet occasionally, until lightly blistered and just starting to soften, 3 to 5 minutes. Transfer to the bowl with the browned butter. Add the Pecorino-Romano, lemon juice, pepper flakes, a pinch of salt and a few grinds of pepper and toss lightly.
Drizzle a second baking sheet with oil. Arrange the chicken cutlets on top in one layer. Sprinkle both sides with salt and pepper. Drizzle the tops of the cutlets with the oil. Broil until just cooked through, 3 to 4 minutes. Divide the chicken and peppers between 4 plates and serve sprinkled with Pecorino-Romano.
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