Heat the Dutch oven on medium-high heat with a tablespoon of butter and the chopped onions. Cook until the onions have turned translucent, about 2-3 minutes. Remove from the Dutch oven and place in a bowl and set aside.
Return the Dutch oven back to the stove and on medium-high heat melt the remaining ¼ cup butter and the flour. Whisk continuously until the flour starts to turn a light colored brown. Continue to whisk and slowly pour in the milk until the flour is evenly dispersed in the milk.
Add the cream, salt, pepper, garlic powder, and chicken broth whisking continuously until it comes to a boil. Reduce the heat to medium low and for 5 minutes. Add the carrots, onion, and broccoli and cook for 15 minutes.
Remove from the heat and add the cheddar cheese and stir until melted.