Melt butter in large stockpot. When a light golden brown, add the flour, sprinkling over butter. Whisking constantly, cook over medium heat about 5 minutes. Slowly add beer continuing to whisk. Slowly add half-n-half, 1/4 cup at a time), followed by stock, continuing to whisk. Bring to a low boil.
Reduce heat to simmer, cook 1-2 minutes, whisking occasionally. Add all spices, cook 1-2 minutes. Add cheese, simmer low for 30-60 minutes. 10-15 minutes before serving, add broccoli to cook al dente.
Serve steaming hot with crackers or crusty bread and a nice cold beer. Enjoy!