Pat the chicken breasts dry with a paper towel. Press your palm down on a chicken breast and slice the chicken in half with a sharp knife parallel to the cutting board. Repeat with the second chicken breast to make a total of 4 cutlets.
Heat the oil in a large heavy-bottomed skillet over medium-high heat until shimmering. Add the capers and fry until golden brown and crispy, 1 to 2 minutes. Remove from the skillet using a slotted spoon and drain on a paper towel-lined plate. Reserve the oil in the skillet and remove from the heat.
Set up a dredging station with 2 shallow bowls or pie plates. Combine the flour with 1/4 cup of the Parmesan, 1 tablespoon kosher salt and the lemon zest in 1 bowl; stir with a fork to combine. Whisk the eggs with 2 tablespoons of the cream in the other bowl.
Heat the reserved oil in the skillet over medium heat.
Working in an assembly line, dredge 1 chicken cutlet in the seasoned flour and turn to coat both sides; tap off any excess. Dip the chicken cutlet into the beaten eggs and coat both sides, letting any excess drip off. Carefully lower the cutlet into the skillet. Repeat with the remaining chicken cutlets, frying them in a single layer, until the chicken is opaque and golden brown, 4 to 5 minutes per side. Transfer the chicken to a paper towel-lined baking sheet and set aside.
Carefully pour off the oil and discard. Wipe out the skillet with a paper towel. Add the butter and cook over medium heat, swirling the skillet occasionally, until the butter bubbles and begins to brown, 1 to 2 minutes. Add the wine and the juice of 1/2 the zested lemon, increase the heat to medium-high and continue to cook, stirring occasionally, until the liquid is reduced and thickened, 5 to 6 minutes.
Add the chicken stock to the skillet and continue to cook over medium-high heat, stirring occasionally, until the sauce is reduced and thickened, about 5 minutes.
Reduce the heat to medium-low and whisk in the remaining 1 tablespoon cream until emulsified.
Taste and adjust the seasoning, if necessary. Return the chicken to the skillet, add the lemon slices and increase the heat to medium. Cook, spooning the sauce over the chicken, until it's warmed through and the lemons have started to soften, 3 to 4 minutes.
Divide the chicken among plates, top with the lemon slices and spoon the sauce over the top.
Sprinkle with the remaining 1/4 cup of Parmesan, the parsley and fried capers and enjoy.