Cut pounded chicken into 8 thin cutlets. Lay out 4 cutlets; top each with 1 slice of ham and 1 slice of cheese, and then top with a second cutlet.
Coat a large nonstick skillet with cooking spray and saute layered chicken-ham-cheese over medium-high heat, carefully turning once, for 8-10 minutes, until golden and cooked through. Remove to a plate and keep warm.
Respray pan and saute mushrooms and shallot over medium-high heat, stirring for 3 minutes. Add wine to deglaze pan, stirring to scrape up any brown bits, until it's reduced by half.
Add chicken broth and tarragon and simmer for 2 minutes. Stir in any chicken juices that have accumulated on plate and add salt and pepper to taste.