Start by rinsing, and chopping the onion, and garlic. Next, cut the broccoli into florets that are about bite-sized pieces, grate the carrots, and shred the cheese. Place them all in bowls and cover them all with plastic wrap. Set the prepared ingredients aside for now.
Next, preheat a small skillet at medium-high heat and then add 1 tablespoon of butter. As soon as the butter melts, add finely chopped garlic. Saute for a minute or until the garlic is toasted.
Then add chopped onion. Saute for a couple of minutes or until the onion starts to turn light golden in color. Make sure to keep mixing from time to time at this high to medium-high temperature. Set this aside.
Add 2 tablespoons of unsalted butter. As soon as the butter melts, add sifted flour and whisk until it mixes with the butter and absorbs completely. This should take about 10 seconds.
Now add half & half and water. Bring this mixture to a boil stirring constantly with a hand whisk to prevent from the flour from settling to the bottom of the pot and then burning.
As soon as it boils reduce the heat to medium-low heat and mix in grated carrots, then add all the spices, namely Mrs. Dash Seasoning, sea salt, cayenne pepper, and ground black pepper (optional, adding it will make the soup a bit spicier).
Next, add in chicken bouillon paste, and sauteed garlic and onion. Stir everything together and let it simmer for 10 minutes stirring every minute or so.
Finally, mix in shredded cheese (it has to be sharp for the best cheddar taste). As soon as it comes to a boil, remove it from the stove and let the soup cool a bit before serving.