Preheat your oven to 350°F.
Slice off the top of each sourdough loaf and hollow out the center, leaving about 1-inch of bread on the sides and bottom. Brush the inside of the bread bowl with olive oil or melted butter and place in the oven for 10 minutes. This helps crisp up the inside of the bread and prevents it from becoming soggy.
In a medium pot, add 4-6 cups of water and 1½ teaspoons of kosher salt and bring to a boil. Add the broccoli florets and cook for 2 minutes. Remove the broccoli florets and plunge them into a bowl of ice water to stop the cooking process. Remove the florets from the ice water and place them on a paper towel to drain.
In a large pot, melt the butter over medium heat. Add the diced onion and jalapeño pepper and sauté until the onion is fragrant and translucent, about 8 minutes. Add the minced garlic and cook for 1 additional minute.
Add the flour and cook for 3 minutes, stirring constantly, to create a roux. Slowly whisk in the broth, ensuring there are no lumps. Add the ½ teaspoon kosher salt and the black pepper. Reduce the heat to low and simmer until thickened, about 20 minutes.
Stir in the milk or heavy cream. Then, gradually add the shredded cheddar cheese, stirring continuously until the cheese melts and is fully incorporated. Reserve 8 broccoli florets and add the rest to the pot. Simmer until heated through, about 5 minutes.
Carefully remove 2 cups of the soup and place in a blender. Don’t fill the blender more than halfway. Work in batches if necessary. Place the lid on the blender, cover with a kitchen towel, and blend on low speed for about 30 seconds. Be careful, the mixture will be very hot.
Add the blended soup back to the pot and stir to combine. Adjust the thickness by adding a little water if necessary. Ladle the hot soup into the prepared sourdough bread bowls. Garnish with a few broccoli florets and extra shredded cheese.
Serve immediately and enjoy!