Place the flour on a plate. In a shallow bowl, beat the eggs; place 1 cup cheese on another plate. Working with 1 cutlet at a time, coat the chicken with the flour, then egg, then cheese, pressing to adhere. Transfer to a plate.
Bring a large pot of water to a boil. Add salt, then the pasta and cook until al dente, 3 minutes. Drain, reserving a ladleful of water.
In a large nonstick skillet, heat 2 tablespoons EVOO, 2 turns of the pan, over medium heat. Add the chicken and cook, turning once, until deeply golden, 10 minutes. Transfer to a platter and cover with foil. Add the remaining 1 tablespoon EVOO, 1 turn of the pan, to the skillet. Add the sliced lemon, garlic and capers and cook for 2 minutes. Stir in the wine and parsley and cook until reduced, 1 minute. Lower the heat and bring to a simmer.
Stir the chicken broth into the sauce for 1 minute. Stir in the butter until melted, then stir in the lemon juice. Spoon a little sauce over the chicken. Add the spinach to the remaining sauce in the skillet and cook until wilted. Add the reserved pasta cooking water. Stir in the pasta and toss to coat; season with salt and pepper. Add the remaining 1/4 cup cheese and toss. Serve the pasta with the chicken.