Rinse and dry the mushrooms, with a sharp knife, cut and remove the legs. Cover the baking dish with parchment and place the hats in it with the holes up. Finely chop the chicken fillet. Heat a pan with 1 tablespoon of vegetable oil over medium heat, add chopped chicken, fry for about 5 minutes, stirring occasionally. As the chicken is fried, finely chop the mushroom legs. Add the chopped mushrooms to the pan with the chicken, and cook together for another 5 minutes, stirring occasionally. Mix the flour with sour cream. Reduce the heat and pour the sour cream into the pan. Salt, season with ground black pepper and mix. Simmer for another 8-10 minutes until the mixture thickens, then remove from heat to cool. While the mixture is cooling, preheat the oven to 180 ° C. Grate feta and mix with chicken and mushrooms. Add finely chopped parsley and mix. Using 2 tablespoons, evenly distribute the filling between the prepared champignon caps. Sprinkle a little grated mozzarella on top of the filling. Bake in the oven for 15-20 minutes, then let the mushrooms cool slightly and serve.