We investigate experimentally the self-phase modulation (SPM) induced by gain dynamics on picosecond pulses in semiconductor optical amplifiers whose gain recovery is enhanced by amplified spontaneous emission (ASE). The observed pulse spectra are highly asymmetric at low drive currents but become more symmetric with increasing current, owing to the ASE-induced reduction in the gain-recovery time down to 9 ps. Furthermore, the amount of spectral broadening is shown to saturate with drive current. We show that a variety of spectral-lobe strengths is selectable, while maintaining a nearly constant small-signal gain, a feature desirable for all-optical signal processing applications.