Prepare the grill for direct cooking over medium heat (180°C to 230°C).
Brush the cooking grates clean. Grill the pepper over direct medium heat, with the lid closed, until the skin is blackened and blistered all over, 12 to 15 minutes, turning occasionally. Place the pepper in a small bowl and cover with plastic wrap to trap the steam. Set aside for at least 10 minutes, then peel the skin from the pepper, discard the stem and seeds and cut into thin strips. Set aside.
Brush the cooking grates clean. Grill the pepper over medium heat, with the lid closed, until the skin is blackened and blistered all over, 12 to 15 minutes, turning occasionally. Place the pepper in a small bowl and cover with plastic wrap to trap the steam. Set aside for at least 10 minutes, then peel the skin from the pepper, discard the stem and seeds and cut into thin strips. Set aside.
Mix the patty ingredients. Then gently form four patties of equal size, each about 10 centimetres wide and 2.5 centimetres thick. With your thumb or the back of a spoon, make a shallow indentation about 2.5 centimetres wide in the centre of the patties to prevent them from forming a dome as they cook. Grill the patties over direct medium heat, with the lid closed, until cooked to medium doneness (70ºC), about 10 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, toast the buns, cut-side down, over direct heat.
Mix the patty ingredients. Then gently form four patties of equal size, each about 10 centimetres wide and 2.5 centimetres thick. With your thumb or the back of a spoon, make a shallow indentation about 2.5 centimetres wide in the centre of the patties to prevent them from forming a dome as they cook. Grill the patties over medium heat, with the lid closed, until cooked to medium doneness (70ºC), about 10 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, toast the buns, cut-side down, over direct heat.
Build each burger on a toasted bun with a patty, grilled peppers, cheese and arugula. Serve warm.
