Preheat a small nonstick skillet over medium-high heat until hot. Add in ½ of the olive oil and swirl to coat. Add the kale, turn the heat to low, and cook stirring constantly until wilted, about 1-2 minutes. Add a pinch of salt to the kale. Transfer the kale to a warm plate.
Return the same pan (which should still be hot) to medium-high heat and add the remaining ½ tbsp olive oil and swirl to coat. Crack in the two eggs and sprinkle them with ⅛ tsp kosher salt. Do not disturb them and allow them to form crispy, crunchy edges. This should take about 1-2 minutes. Be sure to turn the heat down to medium.
You will also need to make the browned butter. To do this, add the butter to a small saucepan and cook over medium heat until the butter turns brown. Keep a close on it as it will only take a few minutes to turn brown. Be sure to swirl the pan occasionally. Set it aside in a bowl. If you leave it in the saucepan it will continue to cook/brown. In the meantime, divide the yogurt between two plates and spread it evenly.
Top the yogurt with half of the kale. Then top the kale with one crispy egg apiece.
Drizzle the salted browned butter over the plates. Serve immediately! Add chopped fresh parsley, if desired.