Heat 1 teaspoon canola oil in a large nonstick skillet over medium heat. Place 2 panini in the pan. Place the medium skillet on top of the panini, then weigh it down with the cans. Cook the panini until golden on one side, about 2 minutes. Reduce the heat to medium-low, flip the panini, replace the top skillet and cans, and cook until the second side is golden, 1 to 3 minutes more. Repeat with another 1 teaspoon oil and the remaining panini.