Combine the balsamic vinegar and brown sugar in a small saucepan over medium heat.

Bring to a gentle simmer, then reduce heat to low and cook for 8-12 minutes, stirring occasionally, until the glaze thickens enough to coat the back of a spoon. It will thicken more as it cools. Remove from heat and set aside.

Preheat oven to 375°F (190°C).

Grease a 9-inch springform pan (or similar round baking dish) and line the bottom with parchment paper. Lightly grease the parchment paper too.

Using a mandoline slicer or a very sharp knife, carefully slice the peeled potatoes into very thin, uniform rounds (about 1/16-inch thick).

In a large mixing bowl, toss the potato slices with 2 tablespoons of olive oil, fresh thyme, garlic powder, salt, and black pepper until evenly coated.

Arrange the potato slices in concentric circles, overlapping slightly, in the prepared springform pan. Start from the outside and work your way inward, creating an even layer. Repeat this process, layering the potatoes until all slices are used. Gently press down on each layer to compact it slightly.

Brush the top layer of potatoes with the remaining 1 tablespoon of olive oil.

Cover the springform pan loosely with aluminum foil.

Bake for 50-60 minutes, or until the potatoes are tender when pierced with a knife.

Remove the foil, increase oven temperature to 400°F (200°C), and continue baking for another 15-20 minutes, or until the top is golden brown and crispy.

Carefully remove the torte from the oven. If using a springform pan, gently release the sides.

Brush the top of the hot potato torte generously with the prepared balsamic glaze. If desired, sprinkle with grated Parmesan cheese.

Allow the torte to rest for 5-10 minutes before slicing into wedges.

Garnish with fresh chopped parsley and serve warm.