Heat a grill pan on the stove. Brush vegetable oil onto prepared sweet mini peppers and place in the grill pan. Grill peppers for about 4 minutes on each side. Remove.
Place slices of bread on a clean work surface. Cover 2 slices of bread with pepper jack cheese. You may need to use more than a slice of cheese for each slice of bread. Layer roasted sweet mini peppers and pastrami on top of cheese. Top with remaining pepper jack cheese and sandwich with remaining slices of bread.
Melt butter in the grill pan. Place sandwiches in the pan and allow it to cook for 2 minutes. Flip and cook for another 2 minutes.