Both methods are correct, but they produce different results. An onion sliced from root to tip, along the grain, holds its shape during cooking and has a milder flavor. An onion sliced across the grain smells sharper because more cells are cut, and it quickly breaks down in the pot and blends into the sauce. Choose based on whether you want the onion to remain visible in the dish.
An onion has a distinct grain that runs from root to tip. When the knife moves parallel to the grain—that is, when cutting from root to tip—most of the cells remain intact; only those along the cut surface are opened. When the knife moves across the grain, many more cells are broken in each slice. That is the whole difference: how many cells are ruptured determines both flavor and texture. That is why the same onion behaves like two different ingredients in the same dish when cut in two different directions. The cutting direction also determines how easily the knife moves through the onion.
An onion sliced along the grain does not break down during cooking. The slices soften in the pan but hold their shape; in kebabs, sautés, baked dishes, and salads, they remain distinct on the plate. The flavor is also milder because the compounds that create sharpness are released when the cells are ruptured. When fewer cells are ruptured, the onion smells milder and is easier to eat raw. Salting and rubbing onions before adding them to piyaz and salads serves the same purpose; it noticeably reduces their sharpness.
An onion sliced across the grain does the opposite. Since the cell walls are cut across, it quickly collapses in the pot, releases its liquid, and melts into the sauce. That is the result you want in soup, meatballs, ground-meat dishes, and sauce bases. The flavor is stronger too; when the same amount of onion is cut this way, it gives the dish noticeably more onion flavor. The sautéing time is also shorter because the cut cells release their liquid faster, allowing the pan to start browning sooner.
In practice, decide by asking two questions. Should the onion be visible in the dish, or should it disappear into it? If you want it visible, slice along the grain; if you want it to disappear, slice across it. The second question is about flavor: do you want a light onion note or a dominant base? You can use both cuts in the same recipe: let some of the onion melt while some stays firm. The same distinction applies to varieties other than yellow onions; green onions and shallots also behave differently depending on how they are cut.
The cutting setup starts the same way for both methods. Cut the onion in half and place the cut side on the board; cutting a round onion while it is whole is difficult as well as dangerous. Leave the root end until last because it holds the layers together. Use a sharp knife; a dull knife crushes rather than cuts the cells, increasing the odor and eye irritation. If you slice an onion soon after taking it out of the refrigerator, it releases less odor and is less likely to slip under the knife. Check for dirt at the root end before discarding it, because that area often remains dirty.