In a small bowl combine egg and the 3 tablespoons milk. In a shallow bowl combine crushed crackers, thyme, salt, paprika, and pepper. Dip chicken pieces, one at a time, in egg mixture; roll in cracker mixture to coat.
In a 12-inch skillet brown chicken in hot oil over medium heat for 10 to 15 minutes, turning occasionally. Drain well.
Add the 1 cup milk to skillet. Heat just to boiling. Reduce heat to medium-low; cover tightly. Cook for 30 minutes. Uncover; cook about 5 to 10 minutes more or until chicken is no longer pink (170 degrees F for breasts; 180 degrees F for thighs and drumsticks). Transfer chicken to a serving platter, reserving drippings for gravy. Cover chicken; keep warm. Prepare Creamy Gravy.
In a screw-top jar combine the 3/4 cup milk, the flour, salt, and black pepper; cover and shake until well combined. Add to reserved drippings in skillet. Stir in an additional 1 cup milk. Cook over medium heat, stirring constantly, until thickened and bubbly. Cook and stir for 1 minute more. (If desired, stir in additional milk to make desired consistency.)