Preheat the oven to 350 degrees F.
Mix the kasha and egg white together in a small bowl until the kasha is well coated. Heat a large nonstick skillet over medium-high heat. Add the kasha mixture and cook, stirring constantly, until the grains soften and separate and the egg adheres to the grains and dries out, about 3 minutes. Let cool completely.
Spread the hazelnuts out on to a baking sheet and bake until golden and toasted, 8 to 10 minutes. Transfer to a cutting board and cool. Rub the skins off the nuts with a kitchen towel, then coarsely chop the nuts.
Bring 2 cups of water to a boil in a medium saucepan. Add the kasha, bring back to a boil, reduce the heat to medium low, cover and simmer until most of the water is absorbed and the kasha is tender, about 15 minutes. Spread the grains out on a baking sheet to cool.
Whisk together the vinegar, oil, honey and 3/4 teaspoon salt in a large bowl. Add the cooled kasha, hazelnuts, carrot, parsley and feta and gently stir to combine. Serve at room temperature.
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