Lay the veal cutlets out on a cutting board and cover with plastic wrap. Use a meat mallet to pound the cutlets into a 1/8 inch even layer. Salt and pepper the veal cutlets liberally on both sides.
Set out a shallow dish or a baking pan and add ½ cup of flour to the pan. One at a time, dredge the veal cutlets in the flour and set aside.
Set a large 14 inch cast-iron skillet on the stovetop. Add the butter and olive oil to the skillet and set the burner on medium heat.
Once the butter has melted, carefully lay four veal cutlets in the hot skillet. Pan fry the cutlets for approximately 1-2 minutes per side, until they are golden brown. Remove the veal cutlets and place them on a paper towel lined baking sheet. Repeat with a second batch. Once all the cutlets are cooked, cover them with foil to keep warm, or place them in a low-temperature oven to keep warm and crispy, around 200°F.
Add the sliced shallots and garlic to the remaining fat in the skillet. Sauté for 2 to 3 minutes to soften, then add in the sliced mushrooms and additional 2 tablespoons of flour. Stir the mushrooms and sauté, so that the mushrooms cook down and a roux is created, about 8 to 10 minutes.
Add the chicken broth, white wine, and dried thyme. Simmer the sauce for 5 to 10 minutes until the sauce thickens. Taste, then salt and pepper as needed. Stir in the chopped parsley and serve.
To serve, place 1 to 2 veal cutlets over fresh cooked pasta or mashed potatoes. Then top with a hearty amount of mushroom sauce. Sprinkle fresh parsley on top.
Notes
Veal Scallopini is best served fresh, but the cutlets do reheat well in an air fryer. Once room temperature, transfer the cutlets and mushroom sauce to separate containers. Keep in the fridge for up to 2 days.