Begin by combining your vegetables, pre-browned sausages, broth, and all seasonings in your slow cooker. Mix thoroughly to ensure the seasonings are evenly distributed throughout.
Set your slow cooker to low and let everything simmer together for 6 hours, allowing the vegetables to become perfectly tender.
When you're approaching the final 30 minutes, combine your milk and cornstarch in a separate bowl until smooth, then stir this mixture into the soup to create that perfect, creamy consistency.
For the finishing touch, turn off the heat and stir in your grated Cheddar cheese until it's completely melted and incorporated.
Serve your creation piping hot, perhaps with a sprinkle of fresh herbs on top for both flavor and presentation.