Cook the bacon in a large Dutch oven over medium-high heat, stirring occasionally, until crisp, about 6 minutes. Transfer to a paper towels-lined plate with a slotted spoon; set aside to drain.
Add the broccoli stems, garlic and onion to the drippings in the pot and cook until slightly softened, 4 to 5 minutes. Add the paprika and crushed red pepper flakes and cook, stirring, until incorporated.
Add the broccoli florets, chicken broth and 2 cups water. Increase the heat to high and bring to a boil. Reduce the heat to medium-low and simmer until thickened and the broccoli stems and florets are cooked through, about 25 minutes.
Remove from the heat and use an immersion blender to blend the soup until creamy. If using a stand blender, work in batches, transferring the soup to the blender and pureeing until smooth (use caution when blending hot liquids); return the soup to the pot.
Return the soup to medium-low heat and whisk in the Cheddar and Monterey Jack until smooth and thickened, about 5 minutes. Season with the vinegar, cayenne and some salt and pepper. Divide among bowls and sprinkle with the bacon and extra Cheddar and Monterey Jack.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.
Copyright 2019 Television Food Network, G.P. All rights reserved.