Preheat the oven to 200 degrees F. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Season the chicken with salt and pepper. Working in batches, cook the chicken until browned and cooked through, about 3 minutes per side. Transfer to a rimmed baking sheet; keep warm in the oven.
Wipe out the skillet and heat the remaining 2 tablespoons olive oil over high heat. Add the shishito peppers and cook, tossing, until browned in spots, 3 minutes. Remove to a plate with a slotted spoon.
Add the garlic and lemon slices to the skillet and cook, stirring occasionally, without flipping the lemons, until they start browning, 2 minutes. Flip the lemons and cook until soft and golden, about 1 minute. Season with 1/2 teaspoon salt and a few grinds of pepper.
Reduce the heat to medium; add the lemon juice and wine and cook until reduced by half, 3 minutes. Add the broth; cook until slightly reduced, 5 to 6 minutes. Remove from the heat; stir in the butter until the sauce thickens. Add the parsley; season with salt and pepper. Serve the chicken with the shishitos and rice; top with the sauce.