My omelette pan is thin, old, and bashed-about. It may once have been non-stick; now the surface is patchy and grimy. You cannot rest the pan flat. It has only one use, for which it is ideal. Omelettes slide about on it, and never get stuck.
It is 20cms, a good size for a two-egg omelette.
The two important points in cooking a tender omelette are to beat the eggs only lightly, and to cook the omelette quickly -- in no more than a minute. Have your grated cheese (no more than a couple of tbsps) and your beaten eggs, seasoned at the last minute (apparently salt does something to the texture) ready.
Warm the pan for 30 seconds on a medium heat, and throw in a knob of butter, which should foam, but not turn brown. (Throughout this process, you may have to move the pan on and off the heat, to regulate the temperature.) When the pan is coated, pour in the eggs. Swirl them around, and with a spatula draw the edges of the omelette as it sets towards the centre of the pan, so that runny egg can fill the place. Keep doing this, to set the omelette as quickly as possible. (You need a very slick pan.)
When you have just a film of runny egg on the surface of the omelette, turn the heat right down, and scatter over the cheese. You do not want too much, or it will get in your way as you try to roll the omelette. This is the bit I am not very good at; but I am getting better. Tip the pan away from you (there should still be a hint of runniness, because the omelette will carry on cooking), and coax the near edge of the omelette to roll over. Keep encouraging it, with a mixture of tipping and pushing. As the omelette falls towards the far edge of the pan, complete the process by turning the pan almost upright to tip the omelette on to a waiting, warm plate.