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 heatproof bowl, being sure to scrape the brown bits from the bottom of the pan.
Spread the shishito peppers in a single layer on a rimmed baking sheet. Broil the peppers, 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 grated Parmesan, lemon juice, a sprinkle of salt and a few grinds of pepper and toss lightly.
Drizzle a second rimmed 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 2 tablespoons oil. Broil until just cooked through, 3 to 4 minutes. Divide the chicken and peppers between 4 plates. Sprinkle with shaved Parmesan and serve with lemon wedges.
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