Preheat the oven to 500° F. Coat dough with 1 tablespoon oil and press onto a rimmed baking sheet in a 1-inch thick layer. Bake until edges are golden and dough is puffed, 8-9 minutes. Remove from oven and reduce heat to 425° F.
Combine kale, beets, remaining oil, salt and pepper in a large bowl. Arrange vegetables evenly over pizza crust, then top with goat cheese. Return to oven and bake until kale is wilted, beets are softened slightly and cheese is melted, 9-10 minutes.
Remove from oven, drizzle with vinegar and top with walnuts.