Prepare the grill for direct cooking over medium-high heat (200º to 260ºC) and preheat a perforated grill pan over direct heat for about 10 minutes.
Mix the ground chuck, 1 tablespoon of the oregano, 1 teaspoon of the salt, 1 teaspoon of the pepper, and the garlic, and then gently form four patties of equal size, each about 18 milliliters 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. Refrigerate the patties until ready to grill.
Combine the onion, bell peppers, 2 tablespoons oil, the remaining ¾ teaspoon salt, the remaining ½ teaspoon pepper, and the crushed red pepper flakes and toss to coat. Spread the vegetables in a single layer on the grill pan and grill over direct medium-high heat, with the lid closed, until tender, about 10 minutes, stirring occasionally. Remove the grill pan and vegetables from the grill and transfer the vegetables to a bowl.
Combine the onion, bell peppers, 2 tablespoons oil, the remaining ¾ teaspoon salt, the remaining ½ teaspoon pepper, and the crushed red pepper flakes and toss to coat. Spread the vegetables in a single layer on the grill pan and grill over medium-high heat, with the lid closed, until tender, about 10 minutes, stirring occasionally. Remove the grill pan and vegetables from the grill and transfer the vegetables to a bowl.
Brush the cut side of the buns with oil. 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 top of each patty to melt, and toast the buns, cut side down, over direct heat.
Brush the cut side of the buns with oil. 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 top of each patty to melt, and toast the buns, cut side down, over direct heat.
Build each cheeseburger on a bun with a patty and onions and bell peppers. Serve warm.
