Prepare the chicken. Make several shallow cuts across the top of each chicken breast without cutting all the way through.
Prepare the baking dish. Preheat the oven to 350°F (180°C). Lightly grease a baking dish with olive oil. Arrange the sliced potatoes in an even layer, then spread the sliced mushrooms over the potatoes.
Make the sauce. Stir together the soy sauce, sour cream, black pepper, and salt in a bowl until smooth.
Coat the chicken. Add the chicken breasts to the sauce and turn them until evenly coated on all sides.
Bake. Place the coated chicken breasts over the potatoes and mushrooms. Bake uncovered for 40 minutes.
Prepare the topping. Mix the shredded mozzarella and cheddar with the egg until well combined.
Add the cheese. Remove the baking dish from the oven and spoon the cheese mixture evenly over the chicken breasts.
Finish baking. Return the dish to the oven and bake for another 20 minutes, until the chicken is cooked through, the potatoes are tender, and the cheese topping is golden. The thickest part of the chicken should reach 165°F (74°C).
Serve. Let the chicken rest for a few minutes before slicing. Serve warm with the potatoes and mushrooms from the baking dish.