In a large, heavy skillet, combine bacon and onions. Cook over medium heat until onions are very soft, and dark brown, about 1 hour. While hot, purée in food processor. Set aside and keep warm. You'll need 1 cup purée for this recipe.
Boil potatoes in salted water until tender. Meanwhile, scald heavy cream (just before it begins to boil). Drain and rice potatoes in a food mill. Note: if you don't have a potato ricer, or a food mill, just use a potato masher. Transfer potatoes to a large bowl, add cream and stir vigorously. Add butter, continuing to stir potatoes Stir in olive oil. Adjust seasoning with salt and white pepper. Set aside and keep warm. You will need 3 cups for this recipe.
Prepare a hot grill. Remove stems and gills from mushrooms. Brush mushrooms with olive oil; season with salt and pepper. Grill 2 to 3 minutes per side. Remove from grill.
Fill mushrooms with bacon puree'. Spoon a good size dollop of mashed potatoes over the bacon. Top with a slice of cheese and a slice of prosciutto, pleated to fit. Place mushrooms on cooler part of grill until cheese is melted and gooey. While mushrooms are warming, season radicchio with salt and pepper. Grill both sides until slightly wilted and starting to turn brown.
Final preparation: In medium bowl, toss grilled radicchio, arugula, olive oil, lemon juice, and garlic. Divide onto four plates. Top each with a stuffed mushroom. Serve immediately.