Place the potatoes in a steamer basket in a large pan over boiling water. Cover the pan and steam until the potatoes are just tender when pierced, 20-30 minutes, depending on size. Let the potatoes cool, then chop them coarsely. Transfer to a large bowl and set aside.
Preheat the oven to 400°F. Coat a 9-inch round cake pan with nonstick cooking spray, then line the bottom with a round of parchment paper, cut to fit. Coat the parchment lightly with nonstick cooking spray as well.
Place a large nonstick skillet over medium-high heat. Add the olive oil, onion, pepper and garlic and cook, stirring occasionally, until the vegetables are tender and wilted, about 10 minutes. Add the walnuts along with 1 tablespoon fresh thyme or 1 teaspoon dried, and cook about 1 minute longer.
Add to the bowl with the potatoes and stir to combine. Season with salt & pepper to taste. Add the egg whites and stir briskly with a fork until evenly mixed.
Press and pat the mixture evenly in the prepared pan. Bake until the potato cake is dry on top and firm to the touch, 30-35 minutes. Remove from the oven and cool 10 minutes.
Slide a table knife around the edge of the pan, then shake the pan to loosen the cake. Invert onto a serving plate, then lift off the pan and peel away the parchment paper.
Meanwhile, stir the yogurt together with the remaining thyme.
Cut the potato cake into 8 wedges and serve warm or at room temperature. Top each wedge with some of the yogurt sauce.