The excitation of large-amplitude plasma waves by intense few-cycle laser pulses in thin overdense plasma layers is studied using one-dimensional particle-in-cell simulation. P-polarized pulses generate relativistic electron pulses (jets) at the irradiated surface that penetrate the layer and then oscillate back and forth due to reflection by self-generated space charge fields building up in front of the surfaces. Counterpropagating plasmons of large amplitude are excited and start to emit radiation at 2ωp and other harmonics of the plasma frequency. The analogy to type III solar radio emission that is driven by electron bursts from deeper layers of the solar corona is pointed out; it highlights the present topic as another example of laboratory astrophysics with lasers.