Dice the carrot and onion into uniform 1/4-inch pieces for even cooking, and cut the broccoli florets into bite-sized pieces. Finely chop the fresh thyme and set aside. Place the hollowed sourdough bread bowls on a baking sheet and toast them in a 350°F oven for 8-10 minutes until they're lightly golden and slightly firm inside—this prevents them from becoming soggy when filled with soup. I like to toast the bread bowls early so they can cool slightly while the soup cooks, making them easier to handle at serving time.

Melt the butter in a large pot over medium heat. Add the diced onion and carrot, stirring occasionally and cooking for about 7 minutes until they're softened and the onion becomes translucent—this builds the flavor foundation. Sprinkle the flour over the vegetables and stir constantly for 2 minutes to create a roux, which will thicken the soup. Gradually pour in the milk while whisking to prevent lumps from forming, ensuring a smooth, creamy base.

Pour the chicken broth into the creamy base and bring to a simmer over medium heat. Add the broccoli florets, bay leaves, cayenne, and a pinch of salt and black pepper. Reduce heat to maintain a gentle simmer and cook for 20 minutes, stirring occasionally, until the broccoli is very tender and the flavors have melded together. The longer simmer allows the vegetables to break down slightly, which will help create a naturally thicker, more cohesive soup when blended.

Remove the pot from heat and fish out the bay leaves with a spoon. Using an immersion blender, blend the soup until it reaches your desired consistency—I prefer a slightly chunky texture with some broccoli pieces remaining, so I blend for just 30-45 seconds rather than until completely smooth. Return the pot to medium-low heat and stir in the sharp cheddar cheese until completely melted and incorporated. Add the chopped fresh thyme and cook for 2-3 minutes more to let the thyme flavor infuse into the soup. Taste and adjust seasoning with additional salt, pepper, or cayenne as needed.

If the soup has thickened too much during cooking, add a splash of water or additional milk and stir well until you reach your preferred consistency. Ladle the hot soup into the toasted bread bowls, filling them generously. Serve immediately while the soup is steaming and the bread bowls are still warm, so the flavors are at their peak.