Step 1In a large skillet, fry the bacon over medium-high heat until crisp, about 10 minutes. Drain the bacon (discard or save the fat for another use) and set aside.
Step 2In a large, heavy-bottom saucepan, melt 2 tablespoons butter over medium-high heat. Add the onions and cook, stirring frequently, until they soften and start to brown, 12 to 15 minutes. Add the white wine and continue to stir, scraping any flavoring from the bottom of the pan.
Step 3When the wine is almost completely absorbed, stir in the greens and cook just until they start to wilt, 2 to 3 minutes. Remove from heat and stir in the vinegar, then taste, and season with one-fourth teaspoon salt and several grinds of black pepper. Stir in the bacon bits, then taste again and adjust the seasoning or vinegar if necessary.
Step 4In a medium bowl, whisk together the eggs and milk, along with the Parmesan cheese. Whisk in a pinch of salt and several grinds of black pepper to season the eggwash. Place the eggwash in a large shallow baking dish or pie plate and set aside.
Step 5Soak the bread slices in the eggwash until just soaked on each side, about 2 minutes per side. Gently spread one-half teaspoon mustard over the one side of each of 4 slices. Onto the other 4 slices, divide the sliced Gruyere, then top with the onion and greens mixture. Sprinkle over the bacon bits, then press the remaining slices of bread (mustard on the inside) onto layered cheese-onions-bacon to form a sandwich.
Step 6Heat the remaining 2 tablespoons butter in a large skillet over medium-high heat until melted and hot.
Step 7Reduce the heat to low and pan-fry the sandwiches, 2 at a time, until crisp and golden and the bread is cooked through, about 5 minutes per side. Cover the skillet while each piece is frying to allow the toast to fully cook and the filling to heat and melt. Adjust the heat as needed to keep each stuffed toast from burning.
Step 8Serve each piece of stuffed toast immediately or hold the finished toast in a warm oven until all of the pieces are fried.