Finely chop the almonds in a mini-chopper or food processor. Transfer the nuts to a shallow dish and stir in the paprika. Lightly whisk the egg whites in another shallow dish. Pat the chicken dry and season both sides with salt and pepper. Dip each piece of chicken into the egg, letting the excess fall back into the dish. Then press both sides of the chicken into the nuts to coat. Place on a baking sheet, cover, and refrigerate for at least 20 minutes to set the crust.
Preheat the oven to 350 degrees F. Place rack on a baking sheet.
Heat a medium nonstick skillet over medium-low heat and add 1 tablespoon of the oil and the butter. Place the chicken smooth-side down in the pan and then raise the heat to medium-high. Cook turning once, until the nuts set and turn golden brown, about 3 to 4 minutes per side. Transfer meat to the prepared pan and bake until an instant-read thermometer inserted into the sides of the chicken registers 160 degrees F, about 20 minutes.
Whisk the vinegar, salt, and pepper to taste in a medium bowl. Gradually whisk in the remaining 3 tablespoons olive oil, starting with a few drops and then adding the rest in a steady stream, to make a smooth dressing. Add the salad greens and toss to coat evenly. Divide the salad evenly among 4 plates, top with the chicken and serve.
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