Heat the olive oil in a large pot over medium heat.
Add the diced onion and sauté for about 5 minutes, or until translucent.
Stir in the minced garlic and thyme, cooking for an additional minute until fragrant.
Add the diced potato and broccoli florets to the pot, followed by the vegetable broth. Bring to a boil.
Once boiling, reduce heat to low, cover, and let simmer for about 15-20 minutes, or until the potatoes and broccoli are tender.
Remove the pot from heat. Use an immersion blender to carefully blend the soup until smooth, or transfer to a standard blender in batches (be cautious with hot liquids).
Return the blended soup to the stove over low heat, and stir in the milk and shredded cheddar cheese until melted and creamy.
Season with salt and pepper to taste, adjusting the consistency with additional milk if desired.
Serve hot, garnished with croutons and chopped chives if desired.
Notes
Optional garnishes include croutons and fresh chives.