Rinse and peel potatoes. Slice thinly (we used a mandolin set to 1/8-inch thickness) and soak slices in cold water to remove excess starch.
Melt butter in a large skillet over low heat and sauté garlic for 1-2 minutes, or until fragrant.
Remove potatoes from water and pat dry with a paper towel.
Place dry potatoes in a large bowl and pour 3/4 garlic butter over them, reserving the rest for later use. Season generously with salt and pepper and toss until everything is evenly coated.
Add oil to the skillet and heat until it’s shimmering and ripply. Remove pan from heat and add potatoes in a spiral formation; the potatoes on the bottom of the skillet will be the top of the “cake.” Once all the slices are in the pan, press it flat like a cake.
Place pan over low heat and cover with a lid for 15 minutes, adding the remaining butter and loosening the edges as the cake cooks. Wipe away moisture that collects on the lid occasionally so the potatoes get crispy as they cook.
Lift edges of cake and check that it is browned; potatoes should be fork tender. Invert cake onto a large plate.
Add more oil to the pan if necessary to keep the cake from sticking. Slide the cake back into the pan with the unbrowned potatoes on the bottom, and cook an additional 15 minutes until golden brown.