Place chicken breasts between 2 sheets of wax paper. Flatten to 1/4 inch thickness, using a meat mallet or rolling pin. Set aside.
Snap off the tough ends of the asparagus. Remove scales from the stalks with a knife or vegetable peeler.
Arrange 1 large or 2 small asparagus spears, one cheese slice, and one sliced mushroom on half of each chicken breast. Top each chicken breast evenly with the chopped artichoke hearts and pimento. Sprinkle evenly with salt and pepper.
Fold the remaining half of each chicken breast over the vegetable mixture; secure with wooden toothpicks. Dredge the chicken in flour; dip in the beaten egg, and coat with the breadcrumbs.
Combine the butter and oil in a large skillet; cook over medium heat until the butter melts. Add the chicken, and cook 7 to 10 minutes on each side or until browned. Remove the chicken from the skillet; drain on paper towels.
Place the chicken on a baking sheet. Top each with 1 remaining cheese slice. Bake at 350 degrees for 15 minutes or until the cheese melts and the chicken is done.