We present new solar phase curve observations of the 2003 EL61 collisional family showing that all the members have light-scattering properties similar to the bright icy satellites of the giant planets and dwarf planets. Compared with other Kuiper Belt objects (KBOs), the five family members we observe (2003 EL61, 2002 TX300, 2003 OP32, 2005 RR43, and 1995 SM55) have conspicuously neutral color (V - I = 0.6-0.8 mag) and flat phase curves at small phase angles (phase coefficients of 0.0-0.1 mag deg-1). Comparing the phase curves we observe for other icy KBOs with the phase curves of icy satellites, we find that the flat phase curves of the 2003 EL61 family are an indication that they have high albedo surfaces coated with fresh ice in the last ~100 Myr. We examine possible resurfacing processes and find none that are plausible. To avoid the influence of cosmic radiation that darkens and reddens most icy surfaces on times scales gsim100 Myr, the family members must be unusually depleted in carbon, or the collision that created the family occurred so recently that the parent body and fragments have not had time to darken. We also find a rotation period of 4.854 (±0.003) h with amplitude 0.26 (±0.04) mag for 2003 OP32.