n a large stock pot or Dutch oven, melt the butter over medium-high heat. Add the sliced leeks and sauté for 5 minutes until leeks are soft. Add the garlic and thyme and cook for 2 more minutes, then add the potatoes, white wine, vegetable broth, and 2 cups of water. Bring to a boil, then reduce heat to a simmer and cook for about 10 minutes until the potatoes are just tender.
While potatoes are cooking, whisk together the remaining 1/2 cup of water with the flour in a small bowl. Whisk the flour-water mixture into the soup once the potatoes are tender and simmer for another 5 minutes.
Remove the soup from heat, remove 2 to 3 cups of the soup from the pot to a blender and blend in the cheese, then slowly stir the cheesy soup mixture back into the soup pot. Season with salt and black pepper and serve hot.