Steps:

• Heat the oven to 400 degrees. Season chicken pieces with salt and pepper. Heat the oil in a large pot or Dutch oven over medium-high heat. In batches, brown the chicken, skin side down to start, about 5 minutes on each side. Remove the chicken to a plate. Add the sausage and sauté until brown, about 3 to 4 minutes. (They will not be cooked all the way through at this point.) Remove with a slotted spoon, cut into 1-inch slices and set aside.
• Remove all but 2 tablespoons oil from the pot and sauté the bell peppers, jalapeños, onion and garlic until soft and beginning to brown, about 5 minutes. 
• Return the sausage to pot, add the cherry peppers, potatoes (if using), chicken broth, wine, vinegar, oregano and season with salt and pepper. Gently mix together, then transfer everything in the pot to a roasting pan. Nestle the chicken in the vegetables and sausage and roast in the oven until chicken is cooked through and sauce has reduced slightly, about 30 minutes.

Steps:

• In a wide, shallow bowl, whisk eggs, milk, salt and pepper until blended. Place the flour in a separate bowl. Line a baking sheet with paper towels.
• In a wide skillet, heat olive and vegetable oils over medium heat until shimmering.
• Working in batches to avoid crowding the pan, lightly dredge the chicken in flour and shake off any excess. Dip into egg batter, let excess batter drip back into the bowl and place in the skillet. Fry, turning once, until golden brown on both sides, about 4 minutes per side. Adjust the heat as the cutlets cook so they brown slowly and evenly, with a steady bubbling. Transfer to the paper-towel-lined pan and repeat with remaining cutlets.
• When all cutlets are browned, remove the pan from the heat and pour off the oil. Wipe out the pan with paper towels. Return the pan to low heat. 
• If making the lemon slices (if not, skip to Step 6 below): Melt 3 tablespoons of the butter and then scatter the lemon slices over the bottom of the pan. Cook, stirring gently occasionally, until the lemon slices are golden and browning around the edges, about 3 minutes. Scoop out the lemon slices and set them aside. 
• Add 3 tablespoons of butter, the wine and lemon juice and bring to a boil. Boil until the liquid is syrupy, 3 to 4 minutes. Pour in the stock, bring to a boil and cook until thickened into a sauce, about 5 minutes. (It will thicken more when you add the cutlets.) Taste and adjust the seasonings with lemon, salt and pepper; it should be quite lemony and not too salty.
• Reduce the heat, tuck the cutlets into the pan and simmer very gently until the sauce is velvety and the chicken pieces are heated through, about 4 minutes. Turn the cutlets over occasionally in the sauce. Place the browned lemon slices on top. Sprinkle with chopped parsley and serve, spooning some of the sauce over each serving.