Preheat the oven to 400 degrees F. Rinse and dry chicken breasts. Place on a baking sheet and season with salt, black pepper, cayenne, and dry thyme leaves. Roast until just cooked through, about 15 minutes. Set aside to cool.
Melt butter in a saute pan on top of the stove. Add celery and onion, with a pinch of salt and pepper, and cook until soft, about 3 minutes. Add garlic and cook 1 minute more. Add flour and stir to coat vegetables well. Cook, stirring often, for about 3 more minutes. Slowly pour in milk, while stirring or whisking. After milk is incorporated, simmer at least 10 minutes.
Cut reserved chicken in large pieces, place in the food processor and pulse until finely chopped. Place in a bowl and pour sauce with vegetables over chicken. Add parsley and mix thoroughly. Make a test croquette by rolling a small amount in bread crumbs, then deep-frying. Adjust seasoning, to taste, with white pepper.
Form chicken mixture into balls, then roll in bread crumbs. Refrigerate and let the croquettes set for at least 1 hour before deep-frying in 325 degree F oil until golden brown. Drain and serve with Creamed Mushrooms.
Heat butter in a saute pan on top of the stove. Add onion with a pinch of salt, pepper, and dry thyme leaves and cook until transparent, about 3 minutes. Add garlic and scallions and cook 1 minute more. Splash with a bit of sherry and cook until evaporated. Stir in flour and cook 1 minute. Slowly stir in milk. Add mushrooms and cook about 20 minutes. Remove from heat and add parsley. Season, to taste, with salt and pepper.