Knead together the butter and flour. Store in the fridge until needed.
Dredge the chicken by gently added to the flour. Shake off any excess then dip in the egg wash. Let the excess drip off and add to the plate.
Add ¼ cup neutral oil and 2 tbs butter to a large skillet. Once hot, cook the chicken in batches until golden and ~85% cooked. Remove to a plate and pour the excess oil into a bowl.
Add 1 tbs olive oil to the same skillet and bring the heat down the medium-low. Add the shallots and garlic. Cook for a few minutes, then lay the lemon slices down. Cook for about 3 minutes per side and then remove to a plate.
Deglaze with the white wine. Cook until halfway reduced, then add the chicken stock and the juice of the rest of the lemon (should be a ½ lemon). Take the butter flour mixture out of the fridge and add to the sauce. Swirl around and bring the mixture to a boil then to a dancing simmer. Let cook until ¼ of the way reduced.
Add the chicken back to the pan, making sure to cover with the sauce. Let cook until the chicken is fully done and the sauce coats the back of a spoon.
Add the parsley and the lemon slices to the chicken and serve.