Bring chicken, 4 cups water, and 1 teaspoon salt to a boil in a Dutch oven. Reduce heat, and simmer 30 minutes or until done. Drain, reserving 3 cups broth. Cover chicken, and keep warm.
Combine crushed garlic and remaining 1/4 teaspoon salt, forming a paste; stir in vinegar.
Melt 1/4 cup butter in a large skillet. Whisk in flour; cook over medium heat, whisking constantly, 1 minute. Gradually whisk in reserved broth and garlic mixture; cook 5 minutes or until smooth and slightly thickened. Gradually whisk 1/2 cup hot broth mixture into eggs; add to remaining hot mixture, whisking constantly. Cook 3 minutes or until mixture reaches 160°. Spoon sauce over chicken.
Stir together 2 tablespoons butter and paprika; drizzle over sauce. Serve with rice or egg noodles.