Rinse the crayfish. Blanch in a large pot with water, peeled onion, salt, pepper and caraway. Remove from the pot.
Rinse the beets and cut into small pieces. Blanch in boiling salted water for 8 minutes, remove and drain. Rinse the scallions and cut off the tough end. Rinse the radishes and remove from greens (saving some for garnish). Blanch the radishes in boiling salted water for 5 minutes, remove and drain.
Break the crayfish and cook the meat with the scallions, radishes, beets and cream. Let simmer. Fry the quail eggs in butter. Peel the apples, core and cut into strips. Add the apples to the ragout and season with salt, pepper and parsley. Serve the stew on plates, garnished with the crayfish heads and radish greens.