Preheat oven to 400°.
Wrap beets individually in foil. Place each foil-wrapped beet on a baking sheet and roast for 1 hour 20 minutes or until softened. Allow to cool completely.
Cook leeks and garlic in 1 tablespoon olive oil in a skillet over medium-high heat for 3-5 minutes or until leeks are softened.
Slice potatoes, apples, and beets thinly to 1/4-inch thickness with a mandoline or knife.
Toss potatoes and apples carefully in a medium bowl with 2 tablespoons olive oil. Toss beets carefully in another bowl with 1 tablespoon olive oil.
Increase oven heat to 425°.
Spread leek mixture evenly over the bottom of a 8-inch cake pan.
Stack potato, apple, and beet slices in individual stacks. Place in pan on top of leeks, splaying out fan-like, one stack after the next to fit. Sprinkle with thyme, rosemary, salt, and pepper.
Cover with foil and bake at 425° for 30 minutes. Uncover and cook for an additional 35 minutes.
Serve topped with goat cheese and walnuts.