Our Solar System was formed from a cloud of gas and dust. Most of the dust mass is contained in amorphous silicates, yet crystalline silicates are abundant throughout the Solar System, reflecting the thermal and chemical alteration of solids during planet formation. (Even primitive bodies such as comets contain crystalline silicates.) Little is known about the evolution of the dust that forms Earth-like planets. Here we report spatially resolved detections and compositional analyses of these building blocks in the innermost two astronomical units of three proto-planetary disks. We find the dust in these regions to be highly crystallized, more so than any other dust observed in young stars until now. In addition, the outer region of one star has equal amounts of pyroxene and olivine, whereas the inner regions are dominated by olivine. The spectral shape of the inner-disk spectra shows surprising similarity with Solar System comets. Radial-mixing models naturally explain this resemblance as well as the gradient in chemical composition. Our observations imply that silicates crystallize before any terrestrial planets are formed, consistent with the composition of meteorites in the Solar System.