The chart below shows the daily mean cloud cover from November through March, and this can be compared to the temperature cross-section that I showed before (see below). Perhaps surprisingly, the sky condition was less cloudy than normal during several of the warmest periods aloft, such as mid-November, early January, and late March, while at other times warm conditions were observed together with clouds. However, the cold spell of late January and early February was relatively clear, which is more what we would expect. It seems clear (pun intended) that some of the most extreme warmth in winter 2014-2015 was brought about by strong ridging (high pressure) aloft, which causes subsidence and therefore clear skies; so we have a potential correlation between warmth aloft and clear skies, which is opposite to the usual association between cold conditions and clear skies in winter.
A time series comparison of the cloud cover anomaly and the surface and 850 mb temperatures (see below) shows much the same thing but also highlights the role of cloud cover in affecting the inversion strength. When cloud cover was below normal, the surface temperature was generally cooler relative to the temperature aloft (e.g. early-mid November, early January, late Jan - early Feb), but when it was cloudy, the surface temperature was generally as warm or even warmer (relative to normal) than the 850 mb temperature (e.g. early December, late December, late Feb - early March). So when looking at the surface temperatures, we can identify the usual warming effect of cloud cover in winter.
It's interesting to look at these relationships in the longer term data. First, the chart below shows monthly mean values of surface temperature anomaly and cloud cover anomaly in winter months since 1998 (when the Fairbanks ASOS came online; note that I looked at this relationship last year over a longer history, but I'm now less confident in the long-term history of cloud cover observations.) The relationship is quite weak and I've labeled November 2014, which was both warm and relatively clear, and January 2012, which was the coldest month in the ASOS era but was also more cloudy than normal.
Looking at 850 mb temperatures rather than surface temperatures, there is very little if any relationship between cloud cover and temperature. Some warm air masses are moist and cloudy, but others are clear because of subsidence aloft.
Now let's look at the relationship between 850 mb and surface temperatures. Clearly the surface temperatures are quite strongly related to temperatures aloft, even in winter when the inversion is generally strong.
What role does cloud cover then play? We can address this by calculating the residual of surface temperature after making a (simple linear) prediction from the 850 mb temperature; see the chart below. The results show that enhanced cloud cover raises the surface temperature relative to where it would normally be based on the 850 mb temperature, and clear skies allow the temperature to be lower than the temperature aloft would suggest. In this regard, November 2014 is no longer an outlier: it was very warm aloft, so it was constrained to be a warm month at the surface, but it was also clear, so it was nearly 5 °F cooler than the 850 mb temperature alone would suggest.
In answer to Gary's query, then, I think we can say that the overall warmth of the lower troposphere (caused by the persistent flow pattern) was the main reason for the mild winter in Fairbanks. Cloud cover played a secondary role in affecting observed surface temperatures and did so by modulating the inversion strength.