Melt butter over medium heat in a large pot. Add onion, carrot, and green onions, and cook, stirring occasionally, until vegetables are tender, about 10 minutes. Add garlic and cook an additional minute.
Stir in flour and dry mustard. Very gradually add broth, stirring after each addition until smooth. Increase heat to medium high and cook until thickened and bubbling.
Add beer and cream, then broccoli and potatoes. Bring to a boil, then reduce heat and simmer, stirring occasionally, until vegetables are tender, about 12 minutes.
Remove from heat and gradually add cheese, stirring until melted. Stir in Worcestershire and Tabasco sauces.