Put oven rack in middle position and preheat oven to 200°F.
Bring broth to a boil in a 2 quart saucepan ver high heat then boil, uncovered, until reduced to about 3/4 cup, about 20 minutes.
Cook shallot in 3 T butter in a 8-10 inch heavy skillet over medium heat, stirring, until shallot begins to turn golden, about 1 minute Add mushrooms, 1 tsp sage, salt and pepper and cook, stirring occasionally until liquid mushrooms give off is evaporated and they begin to brown, 6-8 minutes. Remove from heat.
Put flour in a wide shallow bowl. Gently pound chicken to 1/4 inch thick between 2 sheets of plastic wrap using the flat side of a meat pounder or a rolling pin.
Pat chicken dry and season with salt and pepper, then dredge in flour, 1 piece at a time, shaking off exess.
Heat 1 T each of oil and butter in a 10-inch heavy skillet over medium-high heat until foam subsides, then saute half of the chicken, turning over once, until golden and just cooked through, about 4 minutes total. Transfer cooked chicken to a large ovenproof platter, arranging in one layer, then put platter in oven to keep warm. Wipe out the skillet and cook remaining chicken in same manner, then transfer to oven, arranging in one layer.
Without cleaning, add 1/2 cup wine to skillet and boil over high heat, stirring and scraping up brown bits, about 30 seconds. Add reduced broth, cream and mushrooms, then simmer, stirring occasionally, until sauce is slightly thickned, 6-8 minutes. Add lemon juice and remaining 2 T wine and 1/2 tsp sage.
Serve chicken on top of rice or pasta with sauce poured over.