Preheat the oven to 375˚. Season the chicken with ½ teaspoon salt, ¼ teaspoon paprika and the garlic powder. Place each piece of chicken between 2 sheets of plastic wrap and pound to ¼ inch thick.
Brush the chicken with 1 tablespoon mustard. Top each piece of chicken with a piece of ham, then place a folded slice of cheese in the middle. Roll up the chicken and secure with a toothpick. Season with salt and pepper.
Put the flour in a shallow dish, beat the eggs in a second dish and put the panko in a third. Season the panko with ½ teaspoon salt and ¼ teaspoon each paprika and pepper. Dredge a chicken roll in flour, tapping off the excess, then dip in the beaten egg. Press in the panko to coat. Repeat with the remaining chicken.
Heat 1/8 inch vegetable oil in a large cast-iron skillet over medium-high heat. Set a rack on a rimmed baking sheet. Working in 2 batches, remove the toothpicks and fry the chicken, turning, until golden, 6 to 8 minutes.
Blot the chicken on paper towels, then place on the rack. Add more oil to the skillet as needed. Once all the chicken is fried, transfer the pan to the oven and bake until the chicken is cooked through, 20 to 25 minutes.
Combine the lemon juice, vinegar, remaining 2 teaspoons mustard, honey, garlic and ¼ teaspoon each salt and pepper in a large bowl. Whisk in the olive oil. Add the greens and toss. Serve the chicken with the salad.