Preheat your oven to a cozy 400°F. While it's warming up, lightly butter a 10-inch cast-iron skillet or a round baking dish. This will help your galette slide out effortlessly later on.
Thinly slice those potatoes into 1/16-inch thick rounds. A mandoline slicer is your best friend here for quick, even cuts, but a sharp knife and a steady hand will do just fine. Keep the slices covered with a damp towel.
Melt the butter in a skillet over medium heat. Add the chopped onion and let it sweat until translucent, about 3 minutes. Stir in the garlic and cook for another minute. Remove from heat and whisk in the olive oil, salt, and pepper.
Transfer the butter mixture to a large mixing bowl. Toss the potato slices in.
Now for the fun part! Arrange the potato slices in the prepared skillet, overlapping them in concentric circles, starting from the outside and working your way in. Each slice should overlap its neighbor by about half. Continue layering until all the potatoes are snug in the skillet. Drizzle any remaining butter over the top.
Slide the skillet into the oven and bake until the edges are crispy and the center is tender, about 60-70 minutes. If the top is browning too quickly, tent some foil over the galette to protect it.
Let the galette cool in the skillet for 15-20 minutes. Run a knife around the edge to loosen it, then cut into wedges and serve.