1. Place bread in a food processor or blender, and pulse until fine crumbs form (about 1 1/2 cups). Stir in cheese, 1/2 teaspoon kosher salt, and 1/2 teaspoon black pepper. Transfer mixture to a shallow bowl. Whisk eggs with 1 tablespoon water in a second shallow bowl.
2. Combine tomatoes, garlic, lemon juice, and 1/4 cup olive oil in food processor or blender; pulse until finely chopped. Season vinaigrette to taste with kosher salt and black pepper. Set aside.
3. Slice chicken breasts in half crosswise; pound to 1/4-inch thickness. Dip chicken into egg, allowing excess to drip off, and dredge in breadcrumb mixture. Heat 2 tablespoons oil in a large nonstick skillet over medium heat; add 4 chicken breast halves, and cook 3 to 4 minutes on each side or until golden brown and cooked through. Repeat with remaining 2 tablespoons oil and remaining chicken.
4. Toss arugula and cucumber with reserved vinaigrette in a large bowl. Serve immediately.