How to Make It
Working with 1 chicken breast at a time, place in a large plastic freezer bag, and gently pound to 1/2-inch thickness using the flat side of a meat mallet or a heavy pan. Season flattened chicken breasts all over with salt and pepper, and set aside.
Whisk together eggs and grated cheese in a shallow bowl. Place flour in another shallow bowl.
Pour oil to a depth of 1/8-inch in a deep large skillet, and heat over medium; set up a large plate lined with paper towels. Working with 2 chicken pieces at a time, lightly dredge chicken in flour, and shake off excess. Dip in egg mixture, letting excess drip off. Place in hot oil, and fry until golden brown and just cooked through, about 4 minutes per side. Transfer to paper-towel-lined plate, and repeat with remaining 2 chicken pieces.
Carefuly discard hot oil and wipe skillet clean. Melt 2 tablespoons of the butter in skillet over medium. Add garlic, and cook, stirring constantly, until fragrant, about 30 seconds. Stir in wine, and bring to a simmer over high. Simmer until reduced by half, about 5 minutes. Stir in chicken broth, and return to a vigorous simmer over high. Simmer until reduced by half, about 10 minutes. Stir in cream, and bring to boil over high. Reduce heat to medium-high, and simmer, stirring occasionally, until sauce is thick enough to coat the back of a spoon, about 10 minutes.
While cream sauce simmers, melt remaining 1 tablespoon butter in a medium skillet over medium. Arrange lemon slices in a single layer on bottom of skillet. Cook, undisturbed, until slices are caramelized around edges, 5 to 8 minutes. Remove from heat, and set aside.
Stir parsley into cream sauce, and reduce heat to medium-low. Add chicken, and gently simmer until chicken pieces are heated through, about 2 minutes. Divide chicken among 4 plates; spoon cream sauce evenly over chicken. Top with caramelized lemon slices, and sprinkle with parsley.