We report the results of molecular line observations with the Atacama Large Millimeter/submillimeter Array (ALMA) towards two peculiar icy objects, which were discovered serendipitously by infrared spectroscopic survey of the Galactic plane with the AKARI satellite. Previous infrared observations have reported that both objects show deep ice and dust absorption features that are often seen in embedded young stellar objects (YSOs) or background stars sitting behind dense clouds, however, they are located neither in known star-forming regions nor in known dense clouds. Their infrared spectral energy distributions (SEDs) show a peak around 5 micron, which are incompatible with existing SED models of typical embedded YSOs. The present ALMA observations have detected compact emission of CO(3-2) and SiO(8-7) at the positions of the icy objects. The observed large column ratios of gas-phase SiO/CO (~10^-3) in both objects, as well as their broad line widths (8-14 km/s), imply that they are associated with shocked gas. Although a large dust extinction (Av ~100 mag) is expected from their deep dust/ice absorption, no dust continuum emission is detected, which would suggest a large beam dilution effect due to their compact source sizes. Their systemic velocities are clearly separated from the surrounding CO clouds, suggesting that they are isolated. The characteristics of their SEDs, the presence of deep dust/ice absorption features, compact source size, and SiO-dominated broad molecular line emission, cannot easily be accounted for by any of known interstellar ice-absorption sources. They may represent a previously unknown type of isolated icy objects.