To make the ketchup, heat the oil in a small nonreactive saucepan over medium heat. Add the garlic and chili flakes and cook for 1 minute. Stir in the ketchup and oregano and cook for 2 minutes. Remove from the heat, add the vinegar and basil and season with salt and pepper. Transfer to a bowl and let cool to room temperature or cover and refrigerate for up to 1 day. Bring to room temperature before serving.
To make the fricos, heat an 8-inch nonstick pan over medium-high heat. Add 1/4 cup of the cheese and cook until light brown on the bottom. Turn over with a heat-proof silicone spatula and continue cooking for another 10 to 15 seconds. Remove to a plate and cool. Wipe out the pan with a paper towel and repeat to make 3 more fricos.
Divide the meat into 4 equal portions (about 6 ounces each). Form each portion loosely into a 3/4-inch-thick burger and make a deep depression the center with your thumb. Season both sides of each burger with salt and pepper. Heat a gas grill to high. Brush the burgers with canola oil and grill until desired doneness is reached.
Spread some of the ketchup on the bottom and tops of each bun and place the burgers on the bottom halves. Add some of the arugula to each burger, top with the fricos, and then add the bun tops. Serve immediately.