The kadhai is a deep, curved, narrow-based pan used across Indian cooking for frying, sauteing and reducing. Its geometry solves several problems that a flat-bottomed pan does not.
The curve concentrates the cooking medium
A round base collects oil into a deep pool at the center, so a small volume of fat becomes deep enough to submerge food.
The same volume spread across a flat skillet would be shallow, forcing the cook either to use much more oil or to fry in a layer that browns unevenly.
For a domestic kitchen where cooking fat is a real cost, that geometry is the difference between deep frying being routine and being an occasion.
Sloped sides create a temperature gradient
Heat concentrates at the base and drops as the sides rise, giving the cook two zones in one vessel without adjusting the burner.
Food can be fried at the bottom and moved up the wall to hold and drain, which is how a single cook manages successive batches.
That gradient also lets a cook fry aromatics hard at the base while the mixture already cooked rests higher up, out of the hottest zone.
Depth contains the violent stages of cooking
Adding wet ingredients to hot fat causes vigorous spattering, and adding whole spices to hot oil sends them jumping.
High curved walls keep both inside the pan, which matters in a kitchen where the alternative is oil across a stove and a wall.
The same depth allows a mixture to be stirred energetically without leaving the vessel, which is essential during a long reduction.
Material determines what it is good at
Cast iron and carbon steel hold heat and are used for frying, where dropping food into the oil must not collapse the temperature.
Thin steel and aluminum respond quickly and suit dishes where the cook wants heat to change the moment the flame changes.
Heavy brass and tin-lined vessels are traditional for milk reduction, because mass spreads heat and prevents the hot spots that scorch milk solids.
American stoves complicate the round base
A round-bottomed kadhai sits unstably on an electric coil or a flat glass top, and contact is poor enough that heating becomes uneven.
Flat-bottomed versions are made specifically for this, trading some of the oil-pooling advantage for stable contact with the burner.
A heavy wok is the nearest common substitute, and a Dutch oven works for reductions where depth and thermal mass matter more than the shape of the base.