Liquid crystal "Blue Phases" (BP) have evolved, in the last years, from a scientific curiosity to emerging materials for new photonic and display applications. They possess attractive features over standard nematic liquid crystals, like submillisecond switching times and polarization- independent optical response. However, BPs still present a number of technical issues that prevent their use in practical applications: their phases are only found in limited temperature ranges, thus requiring stabilization of the layers; stabilized BP layers are inhomogeneous and not uniformly oriented, which worsen the optical performance of the devices. It would be essential for practical uses to obtain perfectly aligned and oriented monodomain BP layers, where the alignment and orientation of the cubic lattice are organized in a single 3D structure. In this work we have obtained virtually perfect monodomain BP layers and used them in devices for polarization independent phase modulation. We demonstrate that, under applied voltage, well aligned and oriented layers generate smoother and higher values of the phase shift than inhomogeneous layers, while preserving polarization independency. All BP devices were successfully stabilized in BPI phase, maintaining the layer monodomain homogeneity at room temperature, covering the entire area of the devices with a unique BP phase.