360{\deg} cameras can capture complete environments in a single shot, which makes 360{\deg} imagery alluring in many computer vision tasks. However, monocular depth estimation remains a challenge for 360{\deg} data, particularly for high resolutions like 2K (2048$\times$1024) that are important for novel-view synthesis and virtual reality applications. Current CNN-based methods do not support such high resolutions due to limited GPU memory. In this work, we propose a flexible framework for monocular depth estimation from high-resolution 360{\deg} images using tangent images. We project the 360{\deg} input image onto a set of tangent planes that produce perspective views, which are suitable for the latest, most accurate state-of-the-art perspective monocular depth estimators. We recombine the individual depth estimates using deformable multi-scale alignment followed by gradient-domain blending to improve the consistency of disparity estimates. The result is a dense, high-resolution 360{\deg} depth map with a high level of detail, also for outdoor scenes which are not supported by existing methods.