Preheat oven to 200˚F (90˚C) and set aside a baking sheet or large oven-proof plate.
In a wide bowl or rimmed plate, combine the flour, salt, paprika, and pepper.
In a large skillet, place the butter and olive oil over medium high heat.
Coat all sides of the chicken breasts well with the flour mixture.
Place the chicken in the heated skillet and cook on each side until golden and cooked through to 165˚F (75˚C). If needed, adjust the heat to medium-medium low. Transfer to baking sheet or plate and place in preheated oven to keep warm.
Place the chopped onion in the pan, adding a little more olive oil or butter if needed. Cook until softened and starting to turn golden.
Stir in the garlic and cook until just fragrant 30 seconds to 1 minute.
Add the sliced mushrooms and cook, stirring occasionally, until softened and golden.
Pour in the Champagne and scrape up any browned bits from the bottom of the pan. Simmer until the liquid is reduced by half.
Reduce heat to low and add the heavy cream. Continue to cook until heated through and thickened slightly
Season to taste with salt, black pepper, and red pepper flakes.
Serve immediately over chicken with Parmesan if desired and fresh parsley.