Caramelize the onions. Add the butter to a large pot and turn the heat to low. Once the butter has melted completely, use a spoon to remove the milk solids (white foam) from the surface and discard. Then turn the heat to medium-high and add the onions. Stirring often, sauté until they're a deep golden color and very soft. This should take about 25 minutes. About halfway through the cooking time, reduce the heat to medium-low. Season to taste with salt and pepper. (Here's How to Season to Taste.) Remove the onions from the pot and add them to a small bowl. What was 3 cups will now be about ½ cup. Set aside. Cook sausage. Squeeze the sausage out of its casing into the same pot, break it apart and sauté until it's cooked through. Set aside.
Set oven and prep pan. Preheat the oven to 375°F, and line a baking sheet with foil. Set aside.
Clean mushrooms. Use a damp paper towel or kitchen towel to wipe the mushroom caps clean. Remove the stems from the mushrooms by wiggling them gently. Then use a small spoon to scrape the gills off the undersides of the mushrooms.
Grill mushrooms. Over a high flame, preheat a stovetop grill. (See notes!) On a large plate, drizzle the mushrooms with olive oil on both sides and season generously with salt and pepper. Place them on the preheated grill — if you don’t hear a sizzle, take them off until it’s hot enough that you do. They will not caramelize properly unless it's hot enough. Grill them until they're nicely browned and tender, 3 or 4 minutes per side. When they're done, place them on the foiled covered baking sheet, round side down.
Assemble. Place enough basil leaves on each mushroom to cover the surface. Divide the cheese evenly among them, sprinkling it directly on top of the basil. Then, dividing them equally, add the caramelized onions, followed by the tomato slices and sausage.
Bake. Place the "pizzas" in the preheated 375°F oven and bake until the cheese is completely melted, about 5 minutes.
Garnish and serve. Finely chop the remaining basil and sprinkle it on top of each one and serve.