Step 1

Melt butter in a large Dutch oven over medium until foamy. Add onion, and cook, stirring often, until softened and translucent, about 4 minutes. Add garlic, and cook, stirring constantly, until fragrant, about 1 minute. Add potato and carrot; cook, stirring often, 1 minute. Increase heat to high; stir in broth, half-and-half, and salt, and bring to a boil. Reduce heat to medium, cover, and simmer until potato and carrot are tender, about 10 minutes.

Step 2

Add broccoli, and continue to simmer, uncovered, until broccoli is tender and bright green, about 5 to 6 minutes. Working in batches, if needed, pour soup into a blender. Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over opening. Process until smooth, about 30 to 45 seconds, and return to Dutch oven.

Step 3

Heat soup over medium-low; gradually add cheese, stirring until completely melted after each addition. Serve soup topped with sprinkled cheese.