Melt 1 tablespoon butter in a pot over medium-high heat, then saute onion, stirring occasionally, until tender, about 5 minutes. Add broccoli, broth, and some salt, and bring to a boil, then reduce heat and simmer covered, until broccoli is tender, but still nicely vibrant green, about 7-10 minutes. Fish out a few broccoli florets after about 5 minutes of cooking, and reserve for the topping.

Working in batches, pour soup into a blender, filling no more than halfway. Holding down lid, with a folded kitchen towel, start blender. Puree until smooth, and pour into a large bowl.

Give your pot a quick rinse and place it back over medium-heat. Add the remaining 2 tablespoons of butter, flour, and cook, stirring for 2 minutes. Gradually add the milk and whisk until nicely incorporated, thick and bubbly, about 3 minutes. Season with a little salt and pepper.

Stir your soup into the bechamel sauce you just made, until well incorporated and slightly thick. Season with salt and pepper if necessary. Ladle into bowls, and garnish with the reserved steamed broccoli florets, shredded cheddar cheese, and chopped bacon.