In a large pot on medium heat add 1 tablespoon of butter. When melted add in diced onion and cook for 2-3 minutes until translucent. Add in garlic and cook another 30 seconds or until fragrant.
Add in remaining butter and melt. Add in flour and whisk to combine cooking for 2-3 minutes. Slowly add in milk continuing to whisk mixture. Then slowly add in broth, continuing to whisk. Cook for 3 minutes or until mixture slightly thickens.
Add in chopped broccoli, shredded carrots and pasta and mix to combine. Bring mixture to boil then reduce heat to low to a simmer. Cook for about 8-10 minutes until pasta is cooked. Turn heat off and add in cheese. Mix all together. Add salt and pepper to taste.