Heat oil and 1 tbsp butter in a large pot over medium-high heat. Add onion, carrots, celery, potatoes, and garlic. Cook, stirring occasionally, for 8–10 minutes until softened. Add cauliflower, thyme, oregano, salt, and pepper. Stir to combine.
Add chicken stock. Stir and bring to a boil, then reduce to a gentle simmer. Cover and cook for 10–15 minutes or until veggies are tender (easily pierced with a fork).
In a separate small pot, melt the remaining 4 tbsp butter over medium heat. Sprinkle flour evenly across the top of the melted butter, then whisk constantly for 1 minute. Very gradually add milk while whisking constantly. Cook for 5–6 minutes, until nicely thickened (do not boil). Remove from heat and stir in cheese, a handful at a time, until melted.
Reduce soup heat to low. Pour the cheese sauce into the soup pot and stir. Taste and adjust seasoning with additional salt and pepper (about ¼ tsp more each or to taste). For a creamier soup, blend 1–2 cups and stir it back into the pot.
Ladle into bowls; top with more cheese and fresh thyme if desired. Serve with toasted, buttered bread. Enjoy!
