Preheat oven to 400 degrees.
Heat olive oil in a large skillet over medium heat.
In a shallow bowl, combine salt, pepper, 2 tsp. Old Bay seasoning and 2 Tbsp. cornstarch. Season chicken breasts on both sides, pressing to adhere. Place chicken into the skillet. Brown chicken on both sides over medium-high heat, 4-5 minutes per side. Remove from heat. If using an oven safe skillet, leave the chicken where it is. Alternately, place chicken breasts into a casserole dish or onto a baking sheet.
In a medium bowl, combine the crab meat, 1 tsp. parsley, egg, Worcestershire sauce, 1/2 tsp. of Old Bay seasoning, lemon juice, Dijon mustard and mayonnaise.
Divide the crab mixture among the four chicken breasts, placing a mound of crab mixture on top of the chicken pieces. Top with grated cheddar cheese. Bake for 20 minutes, until chicken reaches an internal temperature of 165 degrees.
While the chicken is baking, make the sauce: In a small saucepan over medium-low heat, whisk 1 1/2 tsp. cornstarch and butter until bubbly. Pour in the milk, 1/2 tsp. Old Bay seasoning and 1 tsp. parsley. Bring to a boil. Lower heat and simmer, whisking occasionally, until chicken is done. Remove from heat. The sauce will thicken as it cools.
Divide the chicken among serving plates and drizzle with Old Bay sauce. Serve warm.