Prick potatoes all over with a fork; place on a microwave-safe plate. Microwave on High, flipping once, until tender and cooked through, 10 to 12 minutes. Let cool for about 5 minutes.
Remove the skin from the potatoes; finely chop the skin and transfer it to a large bowl. Mash the potato flesh using a potato masher (or press through a ricer); add to the bowl. Stir in ½ cup Parmesan, 2 tablespoons mochiko, 1 tablespoon potato starch, 2 teaspoons rosemary and 1 teaspoon garlic powder; knead the mixture until a dough forms.
Add oil to a depth of 1 inch in a Dutch oven or other heavy-bottomed pot. Heat over medium heat to 350°F. (Alternatively, see air-fryer instructions below.)
Meanwhile, shape dough into a (1½-by-11-inch) log; slice into ½-inch-thick pieces (about 20 pieces). Add one-third of the potato pieces (about 7) to the hot oil; cook, flipping often and making sure they don't stick together, until golden brown, 4 to 5 minutes. Transfer to a paper-towel-lined plate; sprinkle with one-third of the ¼ teaspoon salt. Repeat the process twice with the remaining potato pieces and salt.
Transfer the fried potatoes to a plate; sprinkle with the remaining 2 tablespoons Parmesan. Garnish with rosemary, if desired.
