In a small pan sauté 1 tablespoon butter and onions over medium heat until onions are translucent. Remove from heat and set aside. In a small sauce pan bring water to a boil, add beef bouillon and whisk until combined. Remove from heat and set aside.
In a deep pan or soup pot melt 1/4 cup butter over medium heat. Slowly whisk in flour until mixture clumps up or forms a “ball”. Add beef bouillon/water mixture and stir to combine. Slowly whisk in half & half and continue to stir until smooth. (If large lumps remain, mash them out with a rubber spatula)
Add broccoli florets and onions and bring soup to a boil. Reduce heat, cover, and allow to simmer 15-20 minutes or until broccoli is tender and easily pierced with a fork. Stir in salt, white pepper, and cheddar cheese. Stir until cheese is melted. Serve hot and top with additional cheese if desired.