Heat oven to 375°F (190°C). Lightly oil a 12-cup muffin tin with a little of the olive oil or line with silicone cups and set aside.
Warm a medium skillet over medium heat and add the remaining teaspoon of olive oil. Add the chopped onion and bell pepper and cook, stirring occasionally, until the onion begins to soften, about 3 minutes.
Toss in the broccoli and mushrooms, sprinkle with a pinch of salt and a light crack of pepper, and continue to sauté until the vegetables are tender and any excess moisture has evaporated, about 4–5 more minutes. Stir in the smoked paprika at the end, then transfer the veggies to a bowl to cool slightly.
In a separate large bowl, whisk together the whole eggs, egg whites, sea salt and black pepper until frothy and uniform in color.
Fold the warm (not piping hot) sautéed vegetables into the egg mixture along with the crumbled feta and chopped parsley. Stir gently to combine so the feta remains in small curds throughout the mix.
Using a 1/3-cup measure or a small ladle, divide the egg-vegetable mixture evenly among the 12 muffin cups. Don’t overfill — each cup should be about three-quarters full.
Bake on the center rack for 18–22 minutes, until the egg cups are set, slightly puffed and a toothpick inserted into the center comes out clean.
Remove the pan from the oven and let the egg cups rest in the tin for 5 minutes. Run a small knife around each cup to loosen, then transfer to a wire rack to cool briefly before serving.