Melt 1-2 Tbsp butter or non-dairy substitute in a medium sized saucepan.
Add onion, celery, and minced garlic. Sauté for about 3 minutes.
Add thyme, salt, and pepper. Sauté another minute or so.
Add chopped broccoli and broth. Mix well and bring to a boil. Turn down the heat and simmer for 10-12 minutes.
Puree the mixture with a hand blender or any other device you have.
In a smaller pan, melt 3 Tbsp butter or non-dairy substitute. Stir in flour, and let bubble lightly.
Stir in milk or non-dairy substitute. Whisk well. Continue to whisk or stir until the mixture thickens.
Add to pureed soup. Mix well and simmer a few minutes before serving.
If you prefer a thinner soup just add more broth to desired consistency.
Top with some fresh grated Parmesan cheese and seasoned croutons. Gluten-free croutons and non-dairy Parmesan cheeses are available in many stores as well.