Place potatoes and 1/3 cup water in a microwave-safe dish. Cover and microwave on high for 6-7 minutes or until potatoes are fork tender. Drain and set aside.
Slice the chicken breasts in the half through the equator to create 4 fillets. Cover with plastic wrap and pound to an even thickness. Pat dry.
Whisk the flour and all chicken seasoning together in a shallow dish. Dredge each breast in the mixture, shake off any excess, then transfer to a dry surface.
Melt the 2 tablespoons butter in 2 tablespoons oil in a large skillet over medium-high heat. Once hot, add chicken and cook for 4-5 minutes per side (depending on thickness) until golden and cooked through. Transfer chicken to a plate, don't wipe out skillet.
Melt 2 tablespoons butter with 1 tablespoon olive oil over medium heat. Increase temperature to medium-high and add half of the mushrooms. Give them a stir to evenly coat in the butter then arrange mushrooms in an even layer. Cook mushrooms for 3 minutes per side or until golden. Once evenly browned, season mushrooms lightly with freshly cracked salt and pepper and give them a stir. Remove mushrooms from skillet and repeat with remaining mushrooms; transfer all mushrooms to a plate.
Reduce heat to medium and add oil if needed to equal about 1 ½ tablespoons. Add potatoes, shallots, garlic and red pepper flakes. Sauté until shallots are softened, about 2 minutes. Reduce heat to low and stir in heavy cream. Mix chicken broth with cornstarch and add to the skillet followed by balsamic, vinegar and all of the sauce seasonings.
Bring sauce to a simmer while scraping up any brown bits on the bottom of the pan; simmer until sauce thickens to desired consistency, stirring often (not too thick). Push potatoes to the side of the pan and whisk in Parmesan until melted.
Stir in mushrooms and add chicken to warm through to soak up the sauce. Garnish with fresh parsley if desired.