Bring water to a boil in two medium sauce pots. Boil potatoes and cauliflower in separate pots until each are fork tender, about 10 minutes; remove from heat and drain. Place cauliflower in a food processor and process until smooth, about 2 minutes. Place potatoes and processed cauliflower into a large mixer with a whip attachment and mix on medium speed for about 1 minute. Add the butter, yogurt, salt, thyme, garlic powder and black pepper. Mix on the stir setting until all ingredients are well-incorporated, about 2 minutes. Serve.