The recently recognized class of ``transitional disk'' systems consists of young stars with optically thick outer disks but inner disks which are mostly devoid of small dust grains. Here we introduce a further class of ``pre-transitional disks'' with significant near-infrared excesses which indicate the presence of an optically thick inner disk separated from an optically thick outer disk; thus, the spectral energy distributions of pre-transitional disks suggest the incipient development of disk gaps rather than inner holes. In UX Tau A, our analysis of the Spitzer IRS spectrum finds that the near-infrared excess is produced by an inner optically thick disk and that a gap of ~56 AU is present. The Spitzer IRS spectrum of LkCa 15 is suggestive of a gap of ~46 AU, confirming previous millimeter imaging. In addition, UX Tau A contains crystalline silicates in its disk at radii >~ 56 AU which poses a challenge to our understanding of the production of this crystalline material. In contrast, LkCa 15's silicates are amorphous and pristine. UX Tau A and LkCa 15 increase our knowledge of the diversity of dust clearing in low-mass star formation.