Season both sides of chicken breasts with 3/4 teaspoon salt and 1/4 teaspoon pepper. In a large saute pan over medium heat, add 1 tablepsoon butter and 1 tablespoon oil. When butter is melted, add chicken breasts and saute 6-7 minutes per side. Remove chicken and keep warm on plater. Add remaining butter and oil to pan. Saute onions until translucent (3-4 minutes). Add garlic and mushrooms to pan, saute for an additional 5 minutes. Sprinkle flour over mushroom mixture and cook for an additional 30 seconds. Deglaze pan with broth and wine, scraping bits off bottom of pan. Stir in thyme and remaining 1/4 teaspoon salt. Return chicken to pan, simmer until sauce thickens and chicken reaches 170 F on an instant-read thermometer (3-4 minutes). Remove from heat, stir in parsley and white truffle oil, if desired. Serve immediately.