Place the ground turkey meat in a large bowl. Add salt, pepper, white wine, Dijon mustard, garlic, grated onions, thyme, and sage and mix together well. Gently add breadcrumbs and mix until combined. (Overworking the meat at this stage will toughen the burger.).
Divide the burger mixture into 8 equal sized amounts. Pat each portion into a 5" round patty that is a little thicker in the center. Place a mound with 1/4 of the cheese at the center of 4 of the patties. Carefully, place the other 4 patties on top and seal the edges well and pat smooth. If your burgers have become quite soft from handling place back in the fridge for 10 minutes.
To prepare the sauce, heat olive oil in a medium skillet over medium heat. Once it become hot, add the vidalia onion and sauté for 5-7 minutes until the onions are soft and have begun to take on a little color. Season with a small amount of salt and pepper to taste.
Add the mushrooms and wine and stir well. Cook for 3 minutes continuing to stir. To enrich the sauce, add the butter and cook until just melted and incorporated into the wine. Cover and turn off the heat until ready to serve.
While the onions are cooking, heat a large skillet until piping hot but not smoking. Add the burgers to the skillet and cook for 2 minutes to create a good seer. Turn burgers and continue cooking another 2 minutes. Add the final 2 tablespoons of white wine to the pan and cook an additional minute or two.
Toast 8 slices of sourdough bread (or 4 split sourdough rolls) in the toaster or broiler. Top each with sweet roasted pepper half. Place burger on top of the red pepper. Top the burger with 1/4 of the onion-mushroom wine sauce and a second piece of sourdough (or roll tops).
Serve with a glass of wine and extra napkins.