In a large pot, add one tbsp. butter. Once melted, add in onion and stir constantly until onions are soft and translucent.
Add in the remaining butter. Once that has melted, add in flour and whisk together constantly over medium heat for about 1-2 minutes.
Add in chicken broth little by little, continuing to whisk constantly. Then add half and half, still whisking as you pour. Continue stirring for about 2 minutes to allow sauce to thicken a bit.
Add the shredded carrots and broccoli and stir with a wooden spoon. Add garlic salt and pepper stirring until combined.
Reduce the heat to medium low and allow to simmer for 10-15 minutes or until the sauce has thickened further and veggies are soft.
At this point, if you like a very thick soup, feel free to blend it. If you like chunky with a little thickness, you're ready to serve!
Add 2 cups of shredded cheese to the soup and stir until melted. Serve in bowls and immediately add a little extra shredded cheese on top so it melts.
Serve with a side of crusty bread for dipping and enjoy!