Your description's a little sketchy but at a guess you had had pommes Paillason. It's a stone classic. Not difficult but exacting. You've got to follow the directions very closely the first few times until you get a feel for what's going on -- then you can be a little more casual. The dangers are over-cooking, under-cooking, scorching the butter (bitter), not heating the butter enough (greasy), and falling apart. Turn off the phone and bring your nose.
3 medium russet potatoes, about 3/4 lb total. Peeled and cut julienne (may be grated or even cut lyonnaise)
(Optional) herbs as desired
Salt and pepper
4 - 6 tbs butter, cut into tbs size pieces and divided
2 tsp oil, divided
8-10" skillet, seasoned carbon steel (best), seasoned cast iron (not as responsive to heat changes, so not quite as good), or non-stick (not as good, the potatoes release too easily).
Peel the potatoes, then grate or cut in julienne. Put the potatoes in a bowl, cover them with water and drain. Cover with water again and allow to soak for between 20 mintues and 2 hours. Drain the potatoes and dry them thoroughly between towels. When they are dry, toss them with salt, pepper and herbs if desired.
Preheat the pan over medium-high heat. When the pan is hot, add 1 tsp of oil. When the oil shimmers, remove the pan from the flame and immediately add 3 tbs of butter. Swirl the pan, to distribute the butter and return it to the flame. When the butter melts and the foam begins to subside, put the potatoes into the pan, and spread them evenly. Press them gently with a spatula or the back of a spoon, to make sure they are (moderately) compressed and of even depth.
By this time the butter should be fragrant (hazelnuts), if not allow an extra minute or two, then reduce the heat to medium so the potatoes will cook through. After about 8 minutes the edges should be well browned, and the galette ready to turn. Remember that "release" and "properly cooked" are nearly synonymous. If the galette resists moving, allow a few minutes.
When it is ready, slide it from the pan on to a plate. Return the pan to the flame, raise the flame medium-high and add the remaining oil and butter exactly as before. When the foam has subsided, return the galette to the pan by inverting it, raw side down. Allow the potatoes to begin cooking at this heat until the butter is fragrant, then reduce to medium as before. The galette will require about five minutes to finish cooking.
Note: It's important to use a very dry potato. Russets are perfect.
Note: You can cook a large galette as above, and serve it cut in wedges, or make smaller single-serving galettes. The amount of butter should be proportional to pan size more than the amount of potato.
Note: Galettes like these are excellent plated as the base for sauced "proteins." A modern-classic presentation.
Note: Re knife technique -- I know what kind of cook's knife you use. Julienne should be as narrow as the spine of the knife between the handle scales, and about as long as the distance between the first and third rivets. If you can't get them down to 2 or 3mm across, the potato will cook too slowly. For a project this size, it's worth using a mandoline or a grater.