Add cauliflower and potatoes to a large pot. Cover with water and add a pinch of salt.
Bring water to a boil and reduce to a simmer until potatoes and cauliflower are fork tender, about 15 minutes. Drain and return potatoes and cauliflower to the pot.
Add yogurt and cheese to the potatoes and use a potato masher or hand mixer to mash and mix until they are well combined.