To make the ketchup, heat the oil in a small nonreactive saucepan over medium heat. Add the garlic and chile flakes and cook for 1 minute. Add the ketchup and oregano and cook for 2 minutes. Remove from the heat and stir in 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, combine the cheese and flour in a bowl. Heat an 8-inch nonstick pan over medium-high heat. Add 3 tablespoons of the cheese mixture 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 to 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 in the center with your thumb. Season both sides of each burger with salt and pepper. Cook the burgers, using the oil.
IF USING A GRILL: Heat a gas grill to high or heat coals in a charcoal grill until they glow bright orange and ash over. Brush the burgers with the oil. Grill the burgers until golden brown and slightly charred on the first side, about 3 minutes for beef and 5 minutes for turkey. Flip over the burgers. Cook beef burgers until golden brown and slightly charred on the second side, 4 minutes for medium rare (3 minutes if topping with cheese; see step 3) or until cooked to desired degree of doneness. Cook turkey burgers until cooked throughout, about 5 minutes on the second side.
IF USING A GRILL PAN: Heat a grill pan over high heat on top of the stove. Cook the burgers as for a grill, above.
IF USING A SAUTE PAN OR GRIDDLE (PREFERABLY CAST IRON): Heat the oil in the pan or griddle over high heat until the oil begins to shimmer. Cook the burgers until golden brown and slightly charred on the first side, about 3 minutes for beef and 5 minutes for turkey. Flip over the burgers. Cook beef burgers until golden brown and slightly charred on the second side, 4 minutes for medium rare (3 minutes if topping with cheese) or until cooked to desired degree of doneness. Cook turkey burgers until cooked throughout, about 5 minutes on the second side.
Place the burgers on the bun bottoms. Add some of the arugula to each burger, top with the fricos, and then add a dollop of the ketchup. Top with the bun tops and serve immediately.
I think the taste and the texture of buns are best when lightly toasted. To toast a bun on a grill, grill pan, or griddle, split the bun open, place it cut side down on the grill, and grill until light golden brown, about 10 seconds. Alternatively, you can place the split bun halves cut side up on a baking sheet and cook them under a preheated broiler until light golden brown, about 30 seconds. Keep a close eye on them and don't let them get too brown or else that texture will take on a life of its own and defeat the purpose of starting with a soft bun.