Season the chicken breasts with salt, pepper, and oregano.
Sprinkle the chicken with flour to cover evenly.
Heat the butter and oil together in a heavy skillet over a medium high setting.
Add the chicken and cook until browned on both sides. Remove from the pan and set aside.
Add the Meyer lemon slices, shallots, and garlic to the pan, Cook for about 3 minutes, turning the lemon slices once to brown evenly on both sides.
Add the chicken breasts back to the skillet, along with the Italian parsley.
In a small bowl, combine the white wine, Meyer lemon juice, and chicken broth. Pour into the pan.
Cook chicken breasts over a medium low heat until they are thoroughly done.
Serve chicken with pan sauce poured on top.