Peel the potatoes and cut them into about 1/2-inch pieces, then place them in a stock pot and add the diced carrots. The similar size helps the vegetables cook evenly and become tender at the same time.
Add the mashed and diced garlic to the pot with the potatoes and carrots. The garlic will mellow as it simmers and blend into the background instead of tasting sharp.
Pour in 3 cups water so the vegetables are just barely covered. Keep this liquid in the pot because it becomes part of the soup base and carries the potato starch.
Stir in 1 tablespoon Better Than Bouillon Chicken Base. The broth should smell savory right away, which tells you the base is dissolving properly.
Add the fresh chopped parsley or dried parsley and 1 teaspoon kosher salt. This starts seasoning the vegetables early so the soup tastes fuller all the way through.
Bring the pot to a boil over high heat, then lower the heat to medium, vent the lid, and simmer for 20 minutes. The vegetables are ready when a fork slides easily through both the potatoes and carrots.
Take the pot off the heat and mash the vegetables to your preferred texture with a potato masher. Leave some chunks for a rustic finish so the soup stays velvety instead of turning pasty.
Melt the butter in a separate pot over medium heat, then whisk in the flour, 1/2 teaspoon kosher salt, and 1/2 teaspoon black pepper. Cook the roux for 1 to 3 minutes until it looks smooth and smells lightly toasty, not raw.
Slowly whisk in the milk, adding about 1 cup at a time until all 4 cups are incorporated. The mixture should stay smooth as it loosens, which is the sign the roux is absorbing the milk without clumping.
Keep stirring over medium heat until the sauce reaches a boil, then cook for 1 minute while stirring and remove it from the heat. Look for steady bubbles from the center and a sauce that coats the whisk lightly.
Pour the white sauce into the mashed potato mixture and stir until the soup is fully combined. The color should shift to a creamy pale yellow and the texture should feel heavier and silkier right away.
Serve the soup warm with cheddar cheese, chopped green onions, and chopped parsley if desired. The hot soup should melt the cheese slightly and soften the onion just enough to take the edge off.
Refrigerate leftovers after cooling and skip the freezer for this soup. The texture stays nicest when it’s gently rewarmed rather than frozen and thawed.