In a large skillet, heat olive oil over medium-high heat. Add onion; saute until softened, about 2 minutes. Add mushrooms and garlic; cook, stirring frequently, until vegetables are softened and lightly browned, 10-15 minutes. Cover and remove from heat; set aside.
Rinse chicken breast halves under cold water; pat dry with paper towels. On a cutting board, cut each piece in half. Place between sheets of waxed paper; using a rolling pin, pound and roll chicken to 1/4 inch thickness.
In a bowl, combine flour, sage, salt and pepper. Dredge chicken pieces in the mixture, shaking off excess.
In a large skillet, heat 1 tablespoon olive oil and 1 tablespoon butter over medium heat. Cook chicken pieces, 4 or 5 at a time, until cooked through and lightly browned on both sides, about 2 minutes per side; add additional oil and butter, if necessary. Transfer chicken to a plate.
Add balsamic vinegar to the skillet; increase heat to medium-high heat and cook, stirring, to loosen any browned bits from the bottom and sides of the skillet, about 1 minute.
Return chicken and pan juices to the skillet; reduce heat to low and turn medallions once or twice to baste them in the sauce.
Add cooked pasta to the skillet with the vegetables, tossing to combine; warm over low heat until heated through, about 4 minutes.
Transfer pasta and vegetables to a platter. Arrange the chicken medallions on the platter next to the pasta and vegetables; garnish with fresh basil leaves and lemon wedges. Serve immediately.