Combine the chicken, onion, bay leaf, thyme, and parsley in a saucepan. Add enough chicken broth to cover and season with salt & pepper. Bring to a boil, cover, and simmer until chicken is just barely cooked, about 10-15 minutes. Bo not overcook. Remove chicken from the broth, drain and let cool, reserving 3/4 cup of the broth.
To make the bechamel, melt the butter in a saucepan, stir in the flour, and cook for a minute. Add gradually the reserved broth and the milk, season with salt, pepper and nutmeg, and continue cooking, stirring constantly, until thickened & smooth. Cool the sauce, stirring occasionally.
Dip the chicken pieces into the sauce, coating each piece completely. Arrange on a platter and chill until the sauce is firm, at least one hour. [May be prepared ahead]
Dip the chicken pieces into the beaten egg and coat well with the bread crumbs. Fry in hot oil (about 385 F) at least 1/2 inch deep until golden on all sides. Or, better, use a deep-fryer. Drain on paper towels. [May be kept warm in 200 F oven up to 20 minutes]