Make the rice. In a medium saucepan, combine the rice, chicken stock, garlic powder, lemon zest, lemon juice, and 2 tablespoons of the butter. Bring to a boil over high heat. Reduce the heat to low, cover, and simmer until the rice is tender and the liquid has absorbed, about 18 minutes.
Stir in the Parmesan, salt, pepper, capers, if using, and the remaining 2 tablespoons of butter.
While the rice is cooking, make the pork. Working with 1 pork chop at a time, place the meat between 2 sheets of parchment paper or plastic wrap. Using a meat mallet or rolling pin, pound each pork chop to ¼-inch thickness. Season on both sides with the salt and pepper.
In a shallow bowl, beat the eggs. Place the panko in a separate shallow bowl or pie plate. Dip each pork chop into the egg, allowing the excess to drip off. Press each pork chop into the panko to coat evenly.
Heat 2 tablespoons of the vegetable oil in a large skillet, over medium-high heat. Once the oil is glistening, add 2 pork chops and cook, undisturbed, until golden brown, about 3 to 4 minutes. Flip the pork chops and repeat until the other side is golden brown, another 3 to 4 minutes. Transfer the pork to a cutting board. Add the remaining 1 tablespoon of oil to the skillet and repeat with the remaining pork chops. Let the cutlets rest for 5 minutes before slicing.
Make the dressing. To a jar with a tight fitting lid, add the lemon zest, lemon juice, sugar, minced garlic, salt, vinegar, mayonnaise, and olive oil. Seal the jar and shake until emulsified, about 30 seconds.
Divide the rice among 4 bowls. Top with the sliced pork cutlets, arugula, and tomatoes. Drizzle with the dressing and serve with a squeeze of lemon juice if desired.