Caramelized Onion, Mushroom, and Swiss Melt

vegetable oil, (divided)

large onion, (sliced into half circles)

baby portabella mushrooms, (sliced)

kosher salt, (divided)

freshly shredded Swiss cheese, (divided)

slices thick of bread (Italian (French, sourdough, or your favorite))

unsalted butter, (room temperature)