How to Make It
Cut turkey breast into 4 (4-ounce) cutlets. Place each cutlet between 2 sheets of plastic wrap; pound with the flat side of a meat mallet until cutlet is 1/4 inch thick. Cut pounded cutlets in half, creating 8 (2-ounce) cutlets.
Stir together 3 quarts water, salt, and sugar in a large saucepan until salt and sugar dissolve, about 2 minutes. Add cutlets; let stand at room temperature 1 hour. Remove cutlets from brine, and pat dry.
Combine crushed matzo, rice flour, garlic powder, marjoram, oregano, and thyme in a food processor; process until herbs are finely chopped, about 1 minute. Transfer mixture to a medium bowl. Whisk eggs in a separate medium bowl until lightly beaten. Dip cutlets in eggs, and dredge in matzo mixture, pressing to adhere. Place coated cutlets on a plate.
Set a wire rack inside a rimmed baking sheet. Pour canola oil to a depth of 1/4 inch in a large cast-iron skillet; heat over medium-high until shimmering. Add 2 coated cutlets; fry until golden brown and cooked through, about 1 minute and 30 seconds per side. Transfer cutlets to wire rack. Repeat process 3 times with remaining 6 cutlets.
Squeeze juice from 1 lemon half over cutlets. Arrange cutlets on a serving platter. Combine arugula, parsley, and olive oil in a large bowl. Squeeze desired amount of juice from a second lemon half over arugula mixture; toss to coat. Serve cutlets with arugula mixture and remaining lemon halves.