Prep the vegetables and cheese: Finely chop the onion, coarsely grate the carrot, mince the garlic, and prepare the broccoli by peeling and dicing the stems and chopping the florets small. Coarsely grate the cheddar and set aside to warm slightly while you start the soup.
Melt the butter in a large pot over medium heat. Add the onion and carrot and cook, stirring, until softened and translucent, 5–7 minutes. Stir in the garlic and cook until fragrant, 30 seconds.
Sprinkle in the flour and cook, stirring constantly, until it loses its raw smell, about 2 minutes.
Gradually whisk in the vegetable broth until smooth, then whisk in the milk. Bring to a gentle simmer over medium heat, stirring often so the bottom doesn’t scorch.
Add the diced broccoli stems and simmer 3 minutes. Stir in the broccoli florets and simmer until all the broccoli is tender but still bright green, 5–7 minutes.
Partially puree for texture: Using an immersion blender, pulse a few times to puree about half the soup, leaving plenty of pieces for bite. (Alternatively, carefully blend about 2 cups in a blender until smooth, then return it to the pot.)
Stir in the heavy cream. Reduce heat to low and season with the salt, black pepper, and nutmeg. Let the soup drop just below a simmer so it’s hot but not boiling.
Take the pot off the heat. Add the grated cheddar in small handfuls, stirring after each addition until fully melted and smooth, 2–3 minutes total. If needed, return to low heat briefly, but do not let it boil.
Taste and adjust seasoning. Ladle into bowls and garnish with a little extra grated cheddar and the sliced chives. Serve hot.