Stir together shallot, chives, tarragon, vinegar, mustard, garlic, and 1/8 teaspoon pepper in a small bowl. Gradually add 3 tablespoons oil, whisking constantly, until blended. Set dressing aside. Place grated onion in center of a clean kitchen towel; twist towel to squeeze liquid from onion. Stir together onion, hash browns, and egg yolk in a medium bowl. Whisk egg white in a small bowl until frothy; stir into potato mixture. Add flour, 1/2 teaspoon salt, and remaining 1/4 teaspoon pepper; stir until combined. Heat 1 tablespoon oil in a large nonstick skillet over medium; swirl to coat. Drop 1/3 cup potato mixture into skillet, and flatten with a spatula. Repeat 3 times to equal 4 pancakes in skillet. Cook until bottoms are crispy, 4 to 5 minutes. Flip pancakes; cook until bottoms are golden brown and crispy, about 2 minutes. Transfer to a plate. (Do not wipe skillet clean.) Repeat with remaining 1 tablespoon oil and potato mixture. Toss together spring mix and dressing in a large bowl; divide salad among 4 plates. Top each potato pancake with 1 1/2 teaspoons apple butter; sprinkle evenly with remaining 1/4 teaspoon salt. Place 2 potato pancakes on each salad.