Prepare the grill for direct cooking over medium-high heat (about 200ºC).
Grill the poblanos over direct medium-high heat, with the lid closed, until blackened and blistered all over, 10 to 15 minutes, turning occasionally. Place the peppers in a bowl and cover with plastic wrap to trap the steam. Let stand for 10 minutes. Remove the poblanos from the bowl and discard the skin, stems, and seeds. Cut each poblano into two pieces that will fit over the patties.
Grill the poblanos over medium-high heat, with the lid closed, until blackened and blistered all over, 10 to 15 minutes, turning occasionally. Place the peppers in a bowl and cover with plastic wrap to trap the steam. Let stand for 10 minutes. Remove the poblanos from the bowl and discard the skin, stems, and seeds. Cut each poblano into two pieces that will fit over the patties.
In a medium bowl combine the ground chuck, oregano, pepper, and 1 teaspoon salt. Gently form four patties of equal size, each about 18 millimeters thick. With your thumb or the back of a spoon, make a shallow indentation about 2.5 centimeters wide in the center of the patties to prevent them from forming a dome as they cook.
Grill the patties over direct medium-high heat, with the lid closed, until cooked to medium doneness (70ºC), 8 to 10 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, place a slice of cheese on each patty to melt, and toast the buns, cut side down, over direct heat.
Grill the patties over medium-high heat, with the lid closed, until cooked to medium doneness (70ºC), 8 to 10 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, place a slice of cheese on each patty to melt, and toast the buns, cut side down, over direct heat.
Spread the top halves of the buns with avocado and season the avocado with salt. Build each cheeseburger with lettuce, a patty, and a piece of the poblano pepper. Serve right away.
