Preheat the oven to 350 degrees F. Grease a 2 quart glass rectangle pan with 1 tablespoon butter. In a large saute pan, over medium-high heat, melt the remaining 2 tablespoons butter, about 1 minute. Add the onions, 1 teaspoon salt, 1/2 teaspoon cayenne, and 12 turns black pepper and saute for 4 minutes. Stir in the mushrooms and saute for 3 minutes. Add the garlic and saute for 1 minute. Remove from the heat and cool. In a mixing bowl, whisk the eggs for 30 seconds. Add the heavy cream, remaining 1 teaspoon salt, remaining 1/8 teaspoon cayenne pepper, Tabasco, and Worcestershire sauce. Whisk the mixture until fully incorporated. Stir in the sauteed mushrooms. Add the bread cubes and mix well. Pour the filling into the prepared pan. Sprinkle the pudding with the grated cheese. Bake for 55 minutes. Remove from the oven and allow to rest for 5 minutes before serving.