Make the corn pesto: Add the pine nuts to a dry skillet over medium-low heat and toast for 2–3 minutes, stirring often, until lightly golden and fragrant. Remove from the heat and let cool slightly.
Add the toasted pine nuts, corn, basil, Parmesan, garlic, and lemon juice to a food processor. Pulse until finely chopped, then slowly add the olive oil and blend until combined but still slightly textured. Season with salt and pepper to taste.
Prepare the tomatoes: Season the green tomato slices with salt and pepper and let them sit for about 10 minutes. Pat dry. Add the flour to one shallow bowl. In a second bowl, whisk the egg with the milk. In a third, combine the cornmeal, panko, and garlic powder.
Coat each tomato slice in the flour, followed by the egg mixture and then the cornmeal mixture. Heat about ¼ inch of neutral oil in a skillet over medium-high heat. Fry the tomatoes for 2–3 minutes per side, or until golden and crisp. Transfer to a wire rack or paper towel-lined plate and season with salt.
Assemble the sandwiches: Lightly dress the arugula with olive oil, lemon juice, salt, and pepper. Spread a generous layer of corn pesto onto the bottom half of each piece of focaccia. Top each with 2 fried green tomato slices and a handful of dressed arugula. Close with the remaining focaccia and serve right away.