Add the onion, potatoes, cauliflower, garlic, vegetable broth, milk and thyme into a large pan.
Bring to a boil then turn down to a simmer and continue cooking until the potatoes and cauliflower are tender (check with a fork or knife - it will take about 15 mins max).
Transfer to a blender or use a stick blender and blend until completely smooth.
Return to the pan and add the horseradish to taste. The quantity you will need depends on how hot the horseradish is and how pungent you want your soup. Keep tasting as you go until it's right for you.