Prepare your ingredients first. Chop the onion, dice the carrots and celery, mince the garlic, and cut the broccoli into small florets. Shred the cheddar and warm the milk slightly in a small saucepan or microwave.

Melt the butter over medium heat in a large Dutch oven. Add the diced onion, carrots, and celery. Cook until the onions are soft and translucent and the carrots begin to lose their raw edge, about 6 to 8 minutes.

Add the minced garlic and cook until fragrant, about 30 seconds to 1 minute.

Sprinkle the flour over the softened vegetables and stir vigorously to coat. Cook for about 2 minutes to eliminate the raw flour taste.

Gradually whisk in the warm vegetable broth, a little at a time, until the mixture is smooth. Bring to a gentle simmer.

Add the diced potato if using, then add the broccoli florets. Reduce heat to medium-low, cover, and simmer until the vegetables are tender, about 12 to 15 minutes.

Decide your texture: use an immersion blender for a chunkier soup or blend more thoroughly for a silky soup.

Return the pot to low heat and stir in the warmed milk and then the cream or half-and-half if using. Heat gently, but do not boil.

Remove the pot from heat and stir in the shredded cheddar cheese until melted. Add the Dijon mustard, nutmeg, and a couple of grinds of black pepper.

Taste and adjust seasoning with kosher salt, more pepper, and lemon juice if desired. Serve hot with garnish of extra shredded cheddar, chopped chives, or croutons.