Place potatoes in a large pot and cover with 1 inch of water. Bring to a boil and cook for 5-8 minutes, or until tender. Drain then return to the hot pan to dry out.
When the potatoes are cool enough to handle, press through a potato ricer into a large bowl. Stir in 3 tablespoons of butter, garlic, salt, and pepper. Adjust seasoning to taste. Set the potatoes aside to cool. When the potatoes are cooled completely, stir in the cheese and scallions.
Pour bread crumbs into a shallow bowl. Shape 1/2 cup portions of the potatoes into patties (cakes) that are about 3 inches in diameter. Gently press each patty into the bread crumbs turning to coat each side. Place on a parchment paper-lined baking sheet or plate and in the refrigerator for 1 hour to set.
Heat a large skillet over medium heat then add 1/4 cup of vegetable oil. When the oil is hot, add half of the potato cakes. Cook for 3 minutes then carefully turn the cakes over and continue cooking for another 3 minutes, until golden and lightly crisp on both sides.
Transfer the cooked potato cakes to a plate and repeat using the remaining 1/4 cup vegetable oil and potato cakes. Serve hot.