Melt the unsalted butter in a large pot over medium heat until shimmering, then add chopped onion and carrots, sautéing until soft and translucent (about 5 minutes).

Sprinkle in the all-purpose flour, stirring constantly for 2 minutes until it turns pale golden and thickens.

Gradually whisk in the chicken broth to prevent lumps, bringing mixture to a gentle simmer. Cook for 5 minutes until it’s silky and slightly thickened.

Lower heat, pour in the half-and-half, stirring until well combined. Simmer gently for 3 minutes.

Meanwhile, steam or blanch the broccoli florets in boiling water for 2 minutes until bright green and crisp-tender. Shock in ice water.

Drain broccoli, chop coarsely if desired, then stir into the creamy soup base. Let it warm through for 1-2 minutes.

Reduce heat to low, gradually sprinkle in the sharp cheddar cheese, stirring until fully melted and silky.

Add minced garlic, a pinch of nutmeg, and salt and black pepper to taste. Stir and adjust seasoning.