Step 1

Melt the butter in a large saucepan or Dutch oven over medium-high heat. Add the onion, carrot and garlic and cook, stirring occasionally, until just tender, about 5 minutes. Whisk in the flour and cook until golden, 3 to 4 minutes, then gradually whisk in the half-and-half until smooth. Add the chicken broth, broccoli, bay leaves, nutmeg, 1/2 teaspoon salt and few grinds of black pepper Bring to a boil, then reduce to medium-low and simmer, uncovered and stirring occasionally, until the broth has thickened and the broccoli is very tender, about 15 minutes.

Step 2

Meanwhile, prick the potato all over with a fork and put on a microwave-safe plate. Microwave on high until tender, flipping halfway through, about 6 minutes. Let cool, then peel. Use a fork or potato masher to mash until crumbly.

Step 3

Reserve 2 cups of the soup in a liquid measuring cup and discard the bay leaves. Let the remaining hot soup cool for 5 minutes or so, then transfer it to a blender, filling no more than halfway. Add the mashed potato. Remove the center cap from the lid and place it on the blender. Cover the lid with a folded kitchen towel to catch splatters, and pulse until very smooth.

Step 4

Stir the pureed soup and reserved 2 cups of soup back into the pot. Whisk in the cheese until melted and smooth. Season with salt and pepper. Ladle into serving bowls and top with more shredded Cheddar.