Melt butter in a large Dutch oven over medium heat. Add onion and cook 6 to 8 minutes until fully soft and translucent. Add garlic and stir for 1 minute until fragrant.
Sprinkle flour over the onion and stir constantly for 1 minute 30 seconds to 2 minutes until the mixture smells nutty and no raw flour scent remains.
Pour in the vegetable broth in a slow stream, whisking constantly until smooth. Add milk and cream, stir to combine, and bring to a gentle simmer over medium heat. Do not boil.
Add broccoli florets and grated carrot. Cook over medium-low heat for 15 to 18 minutes, stirring occasionally, until the broccoli is fork-tender.
Stir in mustard powder, smoked paprika, and nutmeg. Season with salt and black pepper to taste.
Remove the pot from the heat completely. Wait 1 full minute, then add the shredded cheddar in three additions, stirring well after each until fully melted and smooth.