Soak mussels in a bowl of cold water for at least 10 minutes; remove from water and scrub with a stiff brush to remove dirt if needed. Remove beard using a sharp knife if needed.
Melt butter with the olive oil in a large stockpot over medium heat; cook and stir garlic and shallots until lightly browned, 2 to 3 minutes. Add 1 cup beef broth, anise-flavored liqueur, jalapeno pepper, and red chile pepper to garlic-shallot mixture; bring to a boil. Reduce heat to medium-low and simmer until peppers are softened, about 5 minutes.
Stir wine, remaining 1 cup beef broth, and basil into broth mixture; simmer.
Whisk light cream and cornstarch together in a bowl until cornstarch is dissolved; add to broth mixture and bring to a boil. Add mussels and lemon juice to boiling broth mixture and cover stockpot with a lid; cook until shells open, about 5 minutes. Discard any shells that didn't open.