For the pork: Place the chops between layers of plastic wrap and use a meat mallet to pound the chops to an even thickness of about 1/4 inch. Season the chops on both sides with salt and pepper. Heat a large skillet over medium-high heat. Add about 1/2 inch of vegetable oil. Spread the flour on a plate and season it with salt and pepper. Break the eggs into a wide, shallow bowl. Season the eggs with salt and pepper and whisk together. One piece at a time, dredge the pork lightly in flour and dip in egg, letting the excess drip back into the bowl. Carefully place the pork in the hot oil and cook until golden on the underside, about 2 minutes. Flip and cook until golden on the second side, 1 to 2 minutes. Drain on a paper towel-lined plate, then transfer to a cooling rack set over a baking sheet and place in the oven to keep warm.
For the peas: Heat a medium skillet over medium heat. Add the olive oil. When the oil is hot, add the shallots and garlic and cook, stirring frequently, until softened, 1 to 2 minutes. Add the peas, lemon zest and lemon juice and toss to coat. Season with salt and pepper and keep warm.
For the beurre blanc: Combine the white wine and shallot in a small saucepan. Bring to simmer and cook until the wine is reduced by half. Add the cream and bring to a boil; let boil 30 seconds to 1 minute to thicken. Season with salt and pepper. Reduce heat to low and begin whisking in the cold butter, a piece or two at a time, until the sauce is thick and glossy (do not let the sauce boil). Whisk in the lemon juice and season with salt and pepper. Keep warm.
Serve the pork with the peas, topped with the beurre blanc and a sprinkle of parsley.