Peel and chop 1 1/2 pounds russet potatoes into 1-inch pieces. Place in a large pot, add enough water to cover the potatoes by 2 inches, and add 1 tablespoon of the kosher salt. Bring to a boil over high heat. Reduce the heat to maintain a simmer and cook until the potatoes are tender and can be easily poked with a paring knife, about 10 minutes.

Meanwhile, grate 3 ounces cheese on the large holes of a box grater. Melt 4 tablespoons unsalted butter on the stovetop or in the microwave. Whisk 1 large egg in a small bowl until beaten.

When the potatoes are ready, drain through a heatproof colander. Place the colander over the pot. When the potatoes are cool enough to handle but still warm, pass the potatoes through a ricer or food mill back into the pot. (Alternatively, drain the potatoes, return to the pot, and mash with a potato masher until smooth.)

Add the cheese, melted butter, beaten egg, 1/2 cup all-purpose flour, remaining 1 1/2 teaspoons kosher salt, and 1/2 teaspoon black pepper, and stir to combine. (If starting with leftover mashed potatoes, omit the butter and add the egg and cheese to 2 1/4 cups mashed potatoes. Add the flour 1 tablespoon at a time until the mixture is stiff enough to form into a disc with your hands. Season with salt and pepper to taste). 

Scoop the potato mixture out with a 1/4-cup measuring cup and form a small disc with your hand. Press the disc until 1/2-inch thick. Place on a baking sheet and repeat with the remaining potato mixture.

Line a second baking sheet with paper towels. Heat 3/4 cup vegetable oil in a large frying pan over medium-high heat until shimmering. Place 5 potato cakes in the pan and cook until deep golden brown, 2 1/2 to 3 minutes per side. Transfer to the paper towels and sprinkle lightly with salt.