This dish definitely qualifies as a Izakaya dish especially as “tsukidashi” 突き出し or “otoushi” お通し, the very first small appetizer in Izakaya which is served without your asking for it as soon as you sit down at the counter. Again you can find extensive discussions about this subject (in Japanese).
First prepare a large bowl of cold water with a small amount of vinegar or lemon juice (acidulated water). Cut the “gobo” root into shorter segments, so that they will fit into your bowl with the acidulated water. Under running water, scrape off the brown skin using the back of your knife exposing the white underneath. Since it will discolor very quickly, place these segments in the acidulated water immediately after the skin is removed. Take out one segment at a time from the acidulated water, cut thin ovals by slicing it diagonally using a sharp heavy knife (the root is fibrous and hard). Spread them overlapping like a stack of cards and then julienne them into match sticks. Immediately put them back in the acidualted water and repeat the process until all are julienned. I usually soak julienned burdock root for 10-15 minutes in the acidulated water (the water will become dark grey), then I wash them in running water and drain. If needed, spread them on paper towels and blot dry. Meanwhile, I peel and cut 1-2 medium size carrots (the amount of carrots to the burdock root is totally up to you) in the same manner as the burdock root and set aside.
In a large saute pan like the one shown below or a Chinese wok on a medium high flame, add 1 tbs of vegitable oil and 1 tsp of dark sesame oil. I add red pepper flakes to the oil (the amount is totally up to you). When the oil is hot, add the burdock root and saute for 1-2 minutes so that all is coated with oil. Add carrots and saute another minute or so. Most recipes use sugar and soy sauce which, to me, will make this dish too salty and too sweet. In stead, I use 3 tbs of mirin, 3 tbs of sake and 2 tsp of soy sauce (I add bit more soy sauce later) and braise (you need to keep the ingredients moving using, in my case, long bamboo kitchen chopsticks as seen below, until most of the liquid has evaporated (10 or more minutes). I taste and add a small amount of soy sauce toward the end if necessary. When the liquid is almost all gone, turn off the flame and mix in white sesame. This dish gets better after one day. Serve at room temperature with extra sesame on the top.