Melt 2 tablespoons butter in a large nonstick skillet over medium-high heat. Add the mushrooms to the skillet and cook until mushrooms have given up most of their moisture and start to brown; remove mushrooms from the skillet and set aside.
In a wide dish such as a pie plate mix together the flour and paprika. Reserve 1 tablespoon of the flour mixture and set aside. Roll the chicken breasts in the flour mixture to coat. Add the remaining 4 tablespoons of butter to the skillet and cook chicken over medium-high heat until nicely browned on both sides and then salt and pepper to taste.
Place the mushrooms back into the skillet and mix in the broth, Marsala wine, 1 tablespoon of reserved flour, and tarragon. Reduce heat to a high simmer, cover, and cook until sauce is thickened and chicken is done. (About 20 minutes – chicken breasts should be at least 165F.)