Preheat oven to 350 degrees F. Place the Camembert in a large mixing bowl. Add the Butterkäse, Tilsiter, sour cream, onions, garlic, chives, paprika, and salt and pepper to taste. Combine with your hands, as your body temperature will melt the cheese and make the mixture smoother. The mixture should be thick, about the texture of cream cheese. If it's too thin, add additional Butterkäse or Tilsiter. If it's too thick, add heavy cream.
Season the portabellas with salt and pepper. Spread the cheese mixture inside the cap and press firmly. Place the flour in a shallow dish. Whisk together the eggs and heavy cream in a second shallow dish. Place the bread crumbs in a third shallow dish. Dredge the stuffed portabellas in the flour, tapping gently to remove excess. Place them in the eggwash and cover completely, shaking gently to remove excess. Place the eggwashed portabellas into the breadcrumbs and coat completely. Repeat the process to double-bread.
Heat the oil and butter in a large oven-proof sauté pan over high heat. Place the portabellas in the pan, stuffed-side-down, and sauté until golden brown, about 2 minutes. Turn and sauté until golden brown. Place the pan in the oven and bake until the mushrooms are tender, about 5 to 10 minutes. Remove from oven and allow to cool slightly, allowing the cheese to rest before slicing.