Step 1

Adjust oven rack to middle position and heat oven to 200 degrees. Cut each chicken breast in half crosswise, then cut thick half in half again horizontally, creating 3 cutlets of similar thickness. Place cutlets between sheets of plastic wrap and gently pound to even ¼-inch thickness. Place cutlets in bowl and toss with salt and pepper. Set aside for 15 minutes.

Step 2

Grate peel from 1 end of lemon to yield 1 teaspoon zest. Cut lemon in half crosswise and juice zested lemon half. Measure out 2 tablespoons juice. Slice remaining (unzested) lemon half into four ¼-inch-thick rounds. Discard end. Cut 3 tablespoons butter into ½-inch cubes and transfer to small bowl. Sprinkle cubes with 1 teaspoon flour and toss until cubes are fully coated and no loose flour remains in bowl; refrigerate until needed.

Step 3

Set wire rack in rimmed baking sheet. Spread remaining flour in shallow dish. In medium bowl, whisk eggs with water. Working with 1 cutlet at a time, dredge cutlets in flour, shaking gently to remove excess. Transfer to prepared rack.

Step 4

Heat olive oil and vegetable oil in 12-inch skillet over medium-high heat until shimmering. Working with 1 cutlet at a time, coat 4 cutlets with egg mixture and gently place in skillet. Reduce heat to medium and cook until golden brown on 1 side, about 2 minutes. Using tongs, flip and cook until golden brown on second side, about 2 minutes. Arrange cutlets on serving platter and place in oven. Repeat with remaining cutlets in 2 batches.

Step 5

Discard oil. Gently wipe out skillet with paper towels. Return skillet to medium heat and add lemon slices and remaining 1 tablespoon butter. Cook, flipping slices occasionally, until lightly browned on both sides, about 3 minutes. Transfer lemon slices to cutting board.

Step 6

Add garlic to now-empty skillet and cook, stirring constantly, until fragrant, about 30 seconds. Stir in wine, scraping up any browned bits, and increase heat to medium-high. Bring to rapid simmer and cook until wine is mostly evaporated, 2 to 3 minutes. Stir in broth and continue to simmer, stirring occasionally, until liquid is reduced to ⅔ cup, 6 to 8 minutes longer.

Step 7

Reduce heat to low and stir in reserved lemon zest and juice. Whisk in floured butter cubes, a few at a time, until sauce is thickened to consistency of heavy cream, 1½ to 2 minutes. Off heat, stir in parsley. Season with salt and pepper to taste. Cut browned lemon slices in half and scatter over cutlets. Spoon sauce over cutlets and serve.