For the ketchup: In a small saucepan, heat the olive oil over medium heat. Add the garlic and red pepper flakes and cook, stirring, until the garlic is softened and fragrant, about 1 minute. Stir in the tomato sauce, vinegar, brown sugar, tomato paste, basil, salt and Worcestershire sauce and bring to a simmer. Lower the heat and continue to cook, stirring occasionally, until thick, about 30 minutes. Remove the garlic, cool to room temperature and refrigerate until ready to use (can be refrigerated for up to 1 week).
For the fries: Place two racks in the upper and middle thirds of the oven and preheat the oven to 425 degrees F. Insert a wire rack into a large baking sheet.
Whisk together the flour, salt and pepper in a large bowl. Lightly beat the eggs in a medium bowl. Combine the panko and the Parmesan in another medium bowl.
Slice off both ends of the eggplant. Cut the eggplant in half lengthwise and slice each half into 1/2-to-3/4-inch thick layers, then cut each layer into 1/2-inch sticks. After all of the fries are cut, drop them into the flour and lightly toss them to coat evenly. Working in batches of a few at a time, dip the floured fries into the egg and then into the panko mixture. Place the fries on the wire rack in the baking sheet, leaving an 1/8-inch space between each fry.
Bake the fries until they are golden brown, about 15 minutes. Season the fries with the remaining salt and serve with Basil Ketchup.