Step 1
Stir beaten eggs and Parmigiano-Reggiano cheese together in a shallow bowl. Stir flour salt and pepper together in a separate bowl. Dredge cutlets in the seasoned flour.
Step 2
Bring a large pot of lightly salted water to a boil. Cook linguine at a boil until tender yet firm to the bite about 11 minutes.
Step 3
Meanwhile heat butter and oil in a 12-inch skillet over medium-high heat. Swirl the skillet once the butter has melted to combine butter and oil. Add lemon slices and cook topped with a splatter screen for 2 minutes per side.
Step 4
Dip floured chicken cutlets into the egg mixture and cook in the same skillet about 4 minutes per side. Be careful not to disturb the cutlets as they brown. Remove browned cutlets to a rack and keep warm. Add garlic and cook about 30 seconds. Pour in chicken broth wine and lemon juice. Bring to a boil and cook until liquid has reduced to about 1/3 cup stirring occasionally about 5 minutes.
Step 5
Remove skillet from heat and cool sauce slightly about 2 minutes. Gradually add half-and-half stirring constantly. Season with salt and pepper. Drain linguine reserving some of the cooking water.
Step 6
Place skillet back on low heat. If the sauce is too thick add cooking liquid from the pasta 1 tablespoon at a time until thinned to your desired consistency. If the sauce is too thin create a paste from the leftover flour and about 1 tablespoon of olive oil. Quickly stir or whisk the paste into the sauce and allow to cook to desired consistency.
Step 7
Return chicken pieces and lemon slices to the sauce to warm through about 2 minutes.
Step 8
Serve each cutlet with two lemon slices and sauce over hot cooked linguine. Garnish with fresh parsley.