Place potato and 1/4 teaspoon salt in a large saucepan; cover with water. Bring to a boil. Reduce heat, and simmer 10 minutes or until tender; drain.
While potatoes cook, melt butter in a large Dutch oven over medium heat. Sauté onion, carrot, and garlic 5 minutes or until tender. Add 4 cups broccoli, broth, 2 cups water, and 1/4 teaspoon salt; bring to a simmer, and cook 10 minutes or until broccoli is tender. Stir in potatoes.
Combine 1 cup broccoli and 2 tablespoons water in a microwave-safe bowl. Cover with plastic wrap; vent. Microwave at HIGH 1 minute or until bright green. Drain.
Process mixture in Dutch oven with a hand-held immersion blender until smooth or place half of broccoli mixture in a blender. Remove center piece of blender lid (to allow steam to escape); secure blender lid on blender. Place a clean towel over opening in blender lid (to avoid splatters). Blend until smooth. Pour into a large bowl. Repeat procedure with remaining broccoli mixture. Return pured mixture to Dutch oven.
Add milk and cheese; cook over low heat 2 minutes, stirring until cheese melts and soup is smooth. Stir in steamed broccoli. Ladle soup into individual bowls.
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This soup is delicious! I followed the recipe exactly, but I did use mostly broccoli stalks instead of florets because I was trying to use them up. And it seemed to yield 12 cups instead of the stated 6 cups. I wonder if the stalks made any difference.