Preheat the oven to 375 degrees F (190 degrees C).
Combine goat cheese and sun-dried tomatoes in a small bowl; set aside. Combine flour, salt, and pepper in a second small bowl and set aside.
Place chicken thighs on a cutting board and layer a piece of plastic wrap on top. Pound thighs slightly with a meat mallet to flatten. Divide goat cheese mixture among all of the flattened chicken thighs; spread evenly down the middle. Starting at the short end, roll up the chicken thighs and secure with toothpicks or kitchen twine.
Dredge chicken rolls in the flour mixture, shaking off any excess flour, and set on a plate. Repeat until all of the rolls have been coated with flour.
Heat 2 tablespoons of butter and olive oil in a skillet over medium-high heat. Place chicken thigh rolls in pan and cook until all sides are browned, about 2 minutes per side. Transfer rolls to an oven-safe dish.
Bake in the preheated oven until chicken is no longer pink in the center and the juices run clear, about 25 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Meanwhile, melt remaining 2 tablespoons butter in the skillet over medium heat. Once butter starts to bubble and turn brown, add chopped sage and swirl the pan until sage becomes fragrant. Pour in broth, wine, and lemon juice. Bring sauce to a simmer and reduce heat to low.
Transfer chicken rolls back to the skillet and cook in the sauce for about 2 minutes, turning frequently to coat. Plate rolls and spoon sauce over top.
