Preheat oven to 325°F. Cook carrots according to package directions, reserving remaining carrots for another use; set aside.
Pat chicken dry. Whisk together egg and 2 tablespoons water in a shallow bowl. Place panko on a plate. Dip cutlets into egg mixture, letting excess drip off. Sprinkle each cutlet with 1/8 teaspoon salt; dredge cutlets in panko to coat.
Heat 1 tablespoon oil in a large nonstick ovenproof skillet over medium-high. Cook chicken until browned, 2 to 3 minutes per side. Transfer skillet to oven; bake at 325°F until cooked through, 6 to 8 minutes.
Meanwhile, heat remaining 1 tablespoon oil in a skillet over medium-high. Add shallot; cook, stirring often, until tender and golden, about 3 minutes. Stir in broth and lemon zest and juice. Bring to a simmer over medium-high; cook, stirring occasionally, until reduced to 1/4 cup, 3 to 4 minutes. Whisk in butter, tarragon, and remaining 1/4 teaspoon salt until smooth. Add carrots; toss to coat. Divide carrots and chicken among 4 plates; spoon sauce over carrots. Garnish with tarragon.