In a shallow bowl, whisk eggs, milk, salt, and pepper. Place flour in another bowl. Line a baking sheet with paper towels.

Heat olive and vegetable oils in a skillet over medium heat.

Dredge each cutlet in flour, shake off excess, then dip in egg batter. Fry in batches, turning once, until golden brown (about 4 min per side). You'll know they're ready when they're light amber. Transfer to paper towel-lined pan.

Discard oil and wipe out skillet.

Melt 3 tablespoon butter over medium-low. Add lemon slices and cook until golden at edges (~3 min). Remove and set aside.

Add remaining butter, wine, and lemon juice. Boil until syrupy (3–4 min). Add chicken stock and boil until reduced and thickened (about 5 min). If you're reducing for a sauce, you're looking for nappe. It coats the back of a spoon, and when you draw your finger through it, it leaves a distinct line.

Reduce heat. Return chicken to pan and simmer gently until heated through and coated in sauce (about 4 min), flipping occasionally. You're looking for a bare simmer: small bubbles breaking the surface, not a rolling boil.

Top with lemon slices and chopped parsley. Serve with sauce spooned over each portion.