Position a rack in the middle of the oven and preheat to 350°F (176°C).

To make the topping: Combine flour, oats, melted butter, walnuts, brown sugar, cinnamon, and salt in a large bowl and mix using your hands until large clumps form. Set aside.

To make the filling: Melt butter in a large 10-inch cast-iron skillet over medium heat. Add beets and cook, stirring often, until fork-tender and lightly browned on the edges, about 12 minutes. Remove skillet from heat and let cool for 10 minutes.

To the cooled skillet, add apples, brown sugar, orange juice, cornstarch, vanilla, cinnamon, nutmeg, and cloves. Toss to thoroughly coat the apples and beets. (If your skillet is too small, transfer the mixture into a large bowl, toss to coat, and pour back into the skillet.)

Distribute prepared topping evenly over beets and apple mixture. Bake until the top is lightly browned and the sides are bubbling, about 30 to 40 minutes. Let cool for 20 minutes.

As the crisp is cooling, make the tangy vanilla goat cheese: Whisk yogurt, goat cheese, sugar, and vanilla extract together in a medium bowl until smooth. Serve alongside crisp and spoon onto individual servings.