In a stockpot, melt the butter over medium heat.

Add the onions and cook until translucent.

Mix in the flour and stir for another minute.

Slowly add the milk and whisk well. Keep stirring/whisking.

Add the carrots.

When it starts to get a little thicker and bubbly, add the cheese.

When the cheese is all incorporated, add the broccoli and turn heat to low.

Cook another 10 minutes or so on low hear, stirring every 1-2 minutes to make sure it doesn't burn.