Mix basil, wine, oil and red onion, in a bowl, to prepare the marinade.
Place 1 whole chicken breast between plastic wrap and pound chicken, gently, with a wooden pounder to form a rectangle. Season with salt and pepper and place in the marinade for 30 minutes. Repeat with remaining chicken.
Heat oil in a skillet and add onion and garlic. Once onion is tender add sliced mushrooms and cook until soft. Stir in dried tomatoes and cook 1 minute more. Add 1 tablespoon potato starch or more, depending on the amount of liquid, diluted in ¼ cup water. Mixture must be thick. Season to taste. Separate half of the mushroom mixture to fill the chicken breasts.
Remove chicken breasts from the marinade and place on a working surface. Spread mushroom filling on each breast and roll like a jelly roll. Tie with a string to secure filling.
Place in a ovenproof previously greased pan and bake in a preheated oven at 400°F (200°C), for 25 minutes or until tender. Place remaining mushroom mixture in a skillet and add chicken broth, milk and ½ cup of the reserved tomato liquid. Bring to a boil and reduce for 10 minutes. Season to taste. Thicken sauce with potato starch if necessary and boil 1 more minute.
Remove chicken rolls from the oven and cool for 5 minutes. Remove strings and cut rolls in ¾ inch (1 ½ cm) diagonal slices.
To serve, place slices in a serving platter and cover with sauce.