Preheat the oven to 375 degrees F (or the temperature according the package directions). Brush a 12-cup muffin pan or 24-cup mini muffin pan with olive oil or melted butter.
Heat the 1 to 2 tablespoons olive oil in a large skillet over medium heat and add the onions. Cook, stirring occasionally, until softened, 3 to 5 minutes. Add the garlic and cook, stirring occasionally, for 1 minute. Add the kale and cook, stirring occasionally, until wilted, about 5 minutes. Season with pepper and a pinch or two of salt. Remove from the heat and let cool.
Transfer to a medium bowl, add the feta, parsley and oregano. Stir to combine, taste and add salt and pepper, to taste. Stir in the egg until combined. Set aside.
Unroll the phyllo sheets and cover with plastic wrap, followed by a damp kitchen towel. (You can use the plastic the sheets were rolled in.) Remove 1 sheet of phyllo and place on a clean work surface. Brush with melted butter or olive oil. Place a second sheet on top and brush with more butter or oil. Repeat until you have a stack of 4 phyllo sheets. Using a 3-inch round cookie or biscuit cutter, cut the stack into rounds. Place 1 stack of phyllo rounds in each cup of the prepared muffin pan, pressing in to form a cup shape. Repeat the process with the remaining phyllo sheets until all the muffin cups are filled. Divide the kale and feta filling equally among the phyllo cups.
Bake until the phyllo is lightly browned and the filling is heated through, 20 to 25 minutes. Remove from the oven and serve the spanakopita cups warm or at room temperature.
Thaw the phyllo in the refrigerator for at least 8 hours, then remove from the refrigerator 1 hour before using to bring to room temperature.