Preheat the oven to 350°F. Set a wire rack over a baking sheet.
Heat 1 inch of oil in a shallow skillet to 375°F.
Place the breadcrumbs in one bowl, the eggs in a second, and the flour in a third. Season each with salt and pepper.
Dip the chicken pieces first into the flour, then into the eggs, and lastly into the breadcrumbs and place on a baking sheet or large plate. Repeat with each piece of chicken.
When the oil is hot, fry the breaded chicken until golden brown, turning once, 4-5 minutes per side. Transfer to the prepared wire rack and cover each piece with a handful of cheese mixture before finishing cooking in the oven until the chicken reaches an internal temperature of 165°F, 10-12 minutes minutes.
Place the lemon and tomatoes cut-side up on a baking sheet. Drizzle with oil and sprinkle with salt and pepper. Roast for 10-15 minutes, until soft. Let cool.
To make the vinaigrette, combine all the ingredients except the oil in a blender, food processor, or immersion blender. You may also do this by hand using a whisk. As the mixture is blending, slowly add the oil to emulsify.
In a large bowl, toss the roasted tomatoes with the arugula, cheese and vinaigrette. Divide the salad and lemon halves between plates and serve with chicken.