1. Make the sauce: Mix the mayonnaise, 1 tablespoon dill, 2 teaspoons mustard powder, the lemon juice and 1 tablespoon water in a small bowl. Season with salt and pepper. Set aside.
2. Combine the breadcrumbs, cheese, the remaining 1/2 teaspoon mustard powder, 1 teaspoon salt and 1/4 teaspoon pepper in a shallow bowl.
3. Beat the eggs in another shallow bowl. Heat the vegetable oil in a large nonstick skillet over medium-high heat. Dip the chicken strips in the beaten egg and then in the breadcrumb mixture, pressing so the coating sticks.
4. Working in batches, add the chicken to the skillet and cook until golden brown and crisp, about 3 minutes per side, adjusting the heat as needed. Transfer to paper towels to drain.
5. Place the greens in a large bowl and toss with the tomatoes, the remaining dill and 3 tablespoons of the dill-mayonnaise sauce.
6. Season with salt and pepper. Divide the chicken and salad among plates and serve with the remaining sauce for dipping.