These open-faced egg-and-mushroom sandwiches taste terrific on their own, but when lobster is in season, it makes a delicious add-in. Spoon 1/4 cup coarsely chopped cooked lobster meat over each serving.
Heat 2 tablespoons butter in medium skillet over medium-high heat until foamy. Add mushrooms and shallot. Season with 1/4 teaspoon salt. Cook until mushrooms are soft and golden brown around edges, about 4 minutes. Transfer to a plate.
Toast bread slices, and spread with butter. Season eggs with salt and pepper. Melt remaining 2 tablespoons butter in skillet over medium-high heat. Cook eggs, stirring with a rubber spatula, until just set. Spoon scrambled eggs, then mushrooms, over toasts. Garnish with basil.