1. Place the quartered potatoes in medium sized pot. Cover with at least an inch of cold water. Add a teaspoon of salt. Bring to a boil, reduce to a simmer, and cook until tender (about 20 minutes).
2. While the potatoes are cooking, melt 4 tablespoons of the butter in a large sauté pan on medium heat. Add the chopped onions. garlic, and carrots. Cook until tender, about 6 to 10 minutes.
3. Add ground beef, peas, and corn to the pan with the onions. Cook until no longer pink. Season with salt and pepper. Add the Worcestershire sauce and vegetable broth. Bring the mixture to a simmer and reduce heat to low. Cook uncovered for 10 minutes, adding more broth if necessary to keep the meat from drying out.
4. When the potatoes are done cooking (a fork can easily pierce), remove them from the pot and place them in a bowl with the remaining 4 tablespoons of butter, creamy horseradish, coconut milk, and yogurt. Mash with a fork or potato masher, and season with salt and pepper to taste. The potatoes should be nice a creamy at this point.
5. Preheat oven to 400°F. Spread the beef, onions, and vegetables (if using) in an even layer in a large deep casserole dish (if there is a lot of extra liquid make sure to drain it out). Spread the mashed potatoes over the top of the ground beef. Rough up the surface of the mashed potatoes with a fork so there are peaks that will get well browned.
6. Place in a 400°F oven and cook until browned and bubbling, about 30 minutes. If necessary, broil for the last few minutes to help the surface of the mashed potatoes brown.