According to Louis Uccellini and Mel Shapiro, extratropical cyclone experts with NOAA, these storms may be undergoing a "seclusion" process that creates an semi-isolated tropical system in the midst of an extratropical cyclone. In the seclusion process, a strong extratropical cyclone draws in warm air from the south, and latent heat of condensation from the cyclone's intense precipitation makes this air even warmer. This extra-warm air spirals into the center of the low and wraps around to the west side, where it is pinched off. As result, one has an isolated "warm core" center where deep convection builds and spiral banding can occur. However, unlike a hurricane, there is no eyewall, and no cloud-free eye created by sinking air (subsidence) in the center. The eye-like feature in an extratropical cyclone has upward moving air, and is merely the center where the surface winds spiral into. Spiral bands of convection can develop in the warm air near the center, mimicking the spiral bands of a hurricane. If these convective bands become intense, subsiding air on the flanks of the bands may create subsidence that warms and dries out the surrounding air, creating cloud-free regions near the center that may give it a more eye-like appearance. Another difference with hurricanes is that the upper-level high pressure system (anticyclone) over the extratropical cyclone is displaced to the northeast (downwind) of the center. In a hurricane, the anticyclone is directly over the eye.
The Blizzard of 2006 developed a distinct eye-like feature when it moved offshore over the warm Gulf Stream waters. The storm was undoubtedly tapping the hurricane's source of energy--latent heat of condensation--at the time the photo in Figure 1 was taken, since we can see evidence of spiral banding occurring neat the center of the storm. As seen in Figure 2, the Sea Surface Temperatures increased sharply from 6 to 12 degrees C (43 to 54 degrees F) near where this eye-like feature developed, right along the edge of the Gulf Stream current. There was plenty of water warm water for the storm to tap into for an extra energy source.