To make the ketchup:

Place a large skillet over medium-high heat and when it is hot, add the onion and cook, stirring, until soft and golden, 7 to 10 minutes. Add the remaining ketchup ingredients, reduce the heat to low and cook, stirring occasionally, until very thick, about 35 minutes.

Set aside to cool. If you want a chunky ketchup, cover and refrigerate.

For smooth, transfer to a food processor fitted with a steel blade and process to the desired texture.

Cover and refrigerate at least overnight and up to 3 weeks. (Yield: about 2 cups)

To make the burgers:

Place the beef on a work surface and divide into 4 balls of equal size. Form each into a patty about ¾ to 1 inch thick by tossing it back and forth between your hands.

To ensure more even cooking, make a ½-inch indentation with your thumb in the center of the burger. Handle the patties as little as possible; do not work more than necessary.

Prepare a grill to medium-high. When the coals are glowing red, after 15 to 20 minutes, cover with the grate. After 5 minutes, use a wire brush to thoroughly clean the grate.

Brush the grate with oil. When the coals are covered with pale gray ash and you can leave your hand 5 inches above the fire for 2 to 3 seconds, the coals are ready.

Sprinkle both sides of the burgers with the salt and pepper. Place the burgers on the grate and grill until well seared on both sides, about 5 minutes for medium rare or 6 minutes for medium.

Transfer to buns or a serving platter and serve immediately.

Alternatively, place a cast iron skillet over high heat and when it is hot but not smoking add the burgers to the dry pan. Cook until well seared on both sides, 8 to 10 minutes for medium rare or 10 to 12 minutes for medium.