Warm 2 tablespoons butter in a large, heavy skillet over medium heat. Add the diced bread and cook, tossing, until toasted and golden brown, about 10 minutes. Season generously with salt and pepper to taste. Set aside.
Warm the remaining butter in a medium saucepan over medium heat. Add the onion and sauté, stirring frequently, until tender and translucent, about 8 minutes. Add the potato, broccoli stems, and stock. Bring to a boil. Reduce the heat to low and simmer 10 minutes. Add the broccoli
florets, and simmer until the broccoli is very tender, about 10 minutes.
Stir in the Asiago. Season with lemon juice and salt and pepper to taste. Remove the pot from the heat. Puree the soup in a blender, adding more stock or water if necessary to yield a smooth consistency. Spoon each serving into a bowl, and float the croutons on top.