Heat olive oil in saucepan over medium heat. Add shallots and garlic; cook and stir until softened, 1 or 2 minutes. Pour in white wine, lobster stock, and tomato sauce. Increase heat to medium-high and bring to a simmer. Add mussels. Cover and cook until mussels open, 4 to 6 minutes. Discard any mussels that don't open.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Rub baguette with duck fat on cut sides. Place on a baking sheet and set under broiler. Broil until browned and crunchy, 1 to 2 minutes.
Divide mussels between 2 serving bowls. Pour broth over mussels. Top with sliced basil. Serve with crunchy baguette.