Whisk flour in a saucepan over low heat. Yes, dry. Whisk continuously and quickly pulling the pan off the heat as necessary. When the flour just begins to take on color and smells lightly toasted remove from the heat and dump the flour in a mixing bowl. Whisk for a few more seconds. Allow to cool for at least a minute.
Whisk in a small splash of milk and whisk until smooth. Continue adding milk in small amounts and whisking until you have poured in 3 cups.
Chop potatos into small chunks. Boil until tender, about 15 minutes. Drain and mash. You may need to add the rest of the milk to make a smooth potato mush.
Melt butter in large heavy-bottomed pan over medium heat. Saute carrot, celery, and onion until lightly brown. About 5 minutes. Add flour-milk mixture, salt, pepper, garlic, and potato mixture to pan. Stir well and add more milk as needed to make a nice thick soup. Once the mixture comes to a boil it may thicken a little more so beware.
Whisk in shredded cheese in handfulls until melted. Now is the time to add the frozen mixed veggies. Cook until heated through, about 5 minutes. Stir frequently to prevent burning on the bottom.