Prepare the Oven and Beets: Preheat your oven to 350 degrees Fahrenheit. Thoroughly scrub and rinse the beets, then pat them dry. Lightly coat each beet with olive oil and wrap them in foil, placing two beets per foil packet. Roast in the oven for 1 to 1 ½ hours, or until the beets are tender and can be easily pierced with a knife.
Cool and Peel: Let the beets cool enough to handle. To avoid staining, consider using rubber gloves while you peel the beets. Once peeled, set them aside.
Toast Walnuts: In a dry frying pan over medium heat, toast the chopped walnuts until they are slightly browned, which will bring out their flavor. Remove from heat and set aside.
Make the Vinaigrette: In a small bowl, whisk together the balsamic vinegar and walnut oil. Season the mixture with salt and pepper according to your taste.
Assemble the Stacks: Slice the beets width-wise into three or four rounds, depending on their size. Layer the beet slices with crumbled goat cheese on a plate or platter.
Add Finishing Touches: Drizzle the balsamic walnut oil vinaigrette over the beet and goat cheese layers. Sprinkle with the toasted walnuts and fresh thyme leaves.
