What type heat transfer is used to fry an egg in a pan and how can you tell

The main type and the dominant one over such a process is the conduction process which is marked with yellow. Conduction heat transfer happens when heat is transferred through solid medium by vibrating its atoms so that the heat energy transfer from one atom to the neighboring atom. This exactly what happens at the pan as it is heated then transfers the heat to the oil then to the egg.
The other way which is has an effect but not significant is the convection heat transfer which happens in liquid medium (ex; air and water) this happens when the hot particles of the liquid that is close to the heat source gets hot and so the volume increases so the density decreases and at last these hot particles rise up and other cold particles take its place and so on.
And this happens, marked in red, at the top layer of the egg in the pan.

Hope this helps.