Pour 2 cups boiling water over the mushrooms and let steep for 20 minutes. Drain the mushrooms and reserve the liquid. Roughly chop the mushrooms.
Put the bread into a small bowl, add 1 cup of the cream, and let soak for 15 minutes; drain and discard cream. Cut half the butter into 16 small chunks and set aside.
Grind the chicken in a food processor until smooth, about 30 seconds. Add soaked bread, egg yolks, and nutmeg and season with salt and pepper. Pulse mixture until well combined, about 1 minute. Transfer chicken mixture to a bowl; set aside.
In a bowl, whip the egg whites with a whisk until foamy; fold into the chicken mixture with a rubber spatula. Divide the chicken into 16 even portions (about 1⁄3 cup each) and form each into an oval cutlet. Tuck a reserved chunk of butter into the center of each cutlet. With your hands, shape the cutlets into 3 1⁄2"-long patties that are 1⁄2" thick, making sure no butter is exposed. Place finished patties on a parchment paper–lined sheet pan and refrigerate.
Heat 2 tbsp. of the oil in a 10" skillet over medium-high heat. Add the onions and cook until soft and transparent, about 5 minutes. Add the chopped mushrooms and cook, stirring occasionally, until browned, about 3 minutes. Sprinkle in the flour and mix well. Pour in the reserved mushroom liquid, leaving any sediment behind, and the remaining cream. Bring to a simmer and cook until thickened, about 10 minutes. Turn off heat and season with salt and pepper. Keep mushroom sauce warm.
Remove cutlets from refrigerator and coat them in the bread crumbs, pressing them to make them adhere. In a 12" skillet, heat the remaining butter and oil over medium heat until it sizzles. Working in batches, fry the cutlets, turning once, until golden brown, 4–5 minutes per side. Transfer to a paper towel–lined plate. Serve the cutlets with the mushroom sauce over the top.