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1
Cut each tofu block lengthwise into 3 equal pieces. Drain on paper towels.
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2
Peel hard-boiled eggs and halve lengthwise. Remove hard-boiled yolks and place in a food processor. Thinly slice whites crosswise and reserve.
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3
Add lemon juice, water, mustard, and 1/4 teaspoon each of salt and pepper to yolks in food processor. With motor running, add 1/2 cup oil in a slow stream, processing until emulsified and smooth and scraping down sides as needed. Transfer to a bowl and stir in shallot, parsley, and capers. Reserve 1/4 cup dressing for serving.
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4
Cook beans, uncovered, in well-salted boiling water until crisp-tender, 4 to 5 minutes. Transfer to an ice bath to stop cooking, then drain well. Transfer to a bowl and toss with dressing (excluding 1/4 reserved dressing for serving). Gently toss with arugula, tarragon, and egg whites. Season with salt and pepper.
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5
Heat remaining 2 tablespoons oil in a 12-inch nonstick skillet over medium heat. Meanwhile, lightly beat remaining 2 eggs in a medium bowl. Pat tofu dry and season both sides with 1/2 teaspoon salt (total). Coat tofu with egg, letting excess drip off, then fry, turning once, until golden and heated through, 8 to 10 minutes. Drain briefly on paper towels, then serve with salad and reserved dressing.
