Melt butter and onion in a large pot over medium heat for about 5 minutes or until the onion becomes translucent and fragrant.
Add in the flour and stir until combined and golden, 1 minute.
Add the vegetable broth one cup at a time, very slowly. Stir with each addition. Add the half n half and bay leaves, while stirring.
Simmer for 20 minutes.
Add the broccoli and simmer for 30 minutes.
Remove bay leaves and add in the cheese and simmer for 5-10 minutes. When the cheese is fully incorporated, the soup is ready to serve.
Serve topped with extra cheese, steamed broccoli, black pepper, and a side of hearty bread.