In a large pot over medium heat, melt the butter.
Add the diced onion and cook, stirring occasionally, until softened and translucent.
Stir in the minced garlic and cook for about 1 minute, just until fragrant.
Sprinkle in the flour and stir constantly for 1–2 minutes, until the mixture becomes slightly golden and smells nutty.
Gradually pour in the chicken broth while whisking to prevent lumps. Once fully combined, slowly whisk in the milk.
Add the chopped broccoli florets and shredded carrots. Stir in the kosher salt, ground pepper, and nutmeg.
Bring the mixture to a simmer, and cook uncovered, stirring occasionally, until the broccoli and carrots are tender, about 24–26 minutes.
Reduce the heat to low. Add the shredded cheddar cheese one cup at a time, stirring after each addition until fully melted and smooth.
Remove from heat and serve warm.