In a large nonstick skillet, heat 1 tablespoon EVOO, 1 turn of the pan, over medium-high heat. Add the pineapple slices and cook, turning once, until golden, about 5 minutes; transfer to a plate. In the same skillet, heat the remaining 1 tablespoon EVOO, 1 turn of the pan, over medium-high heat. Add the patties and cook, turning once, until cooked through, 8 to 10 minutes total.