Heat 2 tablespoons of the oil in a nonstick skillet over medium-high heat. Season the tilapia with ½ teaspoon salt and ¼ teaspoon pepper. Cook in batches until cooked through, 2 to 3 minutes per side (adding more oil to the skillet if necessary). Let cool, then break into pieces.
Meanwhile, in a bowl, whisk the lime juice, mustard, honey, the remaining oil, and ¼ teaspoon each salt and pepper. Add the arugula, endive, apple, and almonds and toss; top with the tilapia.
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