Add 1 tablespoon of butter to a dutch oven and let melt over medium heat. Add chopped onions and cook for 2-3 minutes. Add garlic and cook until fragrant, about 1 minute.
Add vegetable broth and milk. Bring to a boil.
Reduce to a simmer over medium-low heat. Add cauliflower florets, potatoes, thyme sprigs (if using), salt, pepper, and poultry seasoning, and stir everything together. Cover and cook for 20 minutes.
Meanwhile, make the grilled cheese by spreading 1/2 tablespoon room temperature butter on one side of each slice of bread.
Place a nonstick skillet over medium heat. Place 1 slice of bread, butter side down, in skillet. Top with 1/2 cup shredded cheddar cheese. Top with a second slice of bread, butter side up. Cook for 3-5 minutes. Then flip the sandwich and cook until the second side is golden brown and cheese is gooey and melted. Repeat this process until you make 3 whole sandwiches.
Cut off the crust from each sandwich and cut each sandwich into bite-size squares. Set aside.
Fish out thyme sprigs from the soup and discard. Take an immersion blender and puree the soup. (see note) Add heavy cream and continue blending until combined.
To serve, ladle the soup into individual bowls, top with grilled cheese croutons and chopped chives.
Notes
If you don't have an immersion blender, ladle the soup into a blender and puree in batches. Don't fill the blender more than 2/3 full. Keep a towel over the top and hold down the lid so the steam doesn't make it fly off. I usually leave the center piece of the lid out to let the steam safely escape. Return blended soup to the pot and proceed.