Heat oil in a soup pan. Chop the bacon into 1 cm cubes and fry until crispy, stirring frequently. Transfer to a paper towel to drain and reserve 1 tbsp of the bacon fat for the soup.
Add the butter and bacon fat to the same pan and saute the onion and the cauliflower until the onion begins to soften and the cauliflower begins to brown a bit on the edges.
Add the potato, thyme and chicken stock. Cover with a lid, reduce the heat and simmer for 20-30 minutes until the vegetables are tender.
Remove from the heat and add the Brie and half of the crispy bacon. Blend with a hand blender until it reaches the desired consistency, thinning with milk or cream, if desired.
Serve garnished with the remaining crispy bacon and some fresh parsley leaves.