We present photometric and spectroscopic observations of the 03fg-like Type Ia supernova (SN Ia) ASASSN-15hy from the ultraviolet (UV) to the near-infrared (NIR). ASASSN-15hy shares many of the hallmark characteristics of 03fg-like SNe Ia, previously referred to as "super-Chandrasekhar"SNe Ia. It is bright in the UV and NIR, lacks a clear i-band secondary maximum, shows a strong and persistent C ii feature, and has a low Si ii λ6355 velocity. However, some of its properties are also extreme among the subgroup. ASASSN-15hy is underluminous (M B,peak = 19.14-0.16+0.11 mag), red ((B-V) {B\max }=0.18-0.03+0.01 mag), yet slowly declining (Δm 15(B) = 0.72 ± 0.04 mag). It has the most delayed onset of the i-band maximum of any 03fg-like SN. ASASSN-15hy lacks the prominent H-band break emission feature that is typically present during the first month past maximum in normal SNe Ia. Such events may be a potential problem for high-redshift SN Ia cosmology. ASASSN-15hy may be explained in the context of an explosion of a degenerate core inside a nondegenerate envelope. The explosion impacting the nondegenerate envelope with a large mass provides additional luminosity and low ejecta velocities. An initial deflagration burning phase is critical in reproducing the low 56Ni mass and luminosity, while the large core mass is essential in providing the large diffusion timescales required to produce the broad light curves. The model consists of a rapidly rotating 1.47 M o˙ degenerate core and a 0.8 M o˙ nondegenerate envelope. This "deflagration core-degenerate"scenario may result from the merger between a white dwarf and the degenerate core of an asymptotic giant branch star.