[Update June 11: and just like clockwork, thunder was reported today at 4:05pm AKDT. Remarkable.]
The chart below shows the dates of all 408 days with "thunderstorm" reported in the hourly observations at Fairbanks since 1950. The ASOS platform was introduced in December 1997, so that could explain the lack of outlier dates in recent years, although the ASOS lightning detector is reported to have a higher rate of reporting than human observers (and a very low false alarm rate).
In my post of 2 years ago I looked at seasonal changes in humidity as a possible explanation for the very rapid increase in thunderstorm frequency in early June, but the conclusion was that humidity alone probably isn't enough to explain the phenomenon.
To take this investigation one step further, I recently calculated a measure of vertical instability by taking the difference in equivalent potential temperature (θe) between the surface and 500 mb. The idea here is that a large vertical gradient (decrease) in θe indicates greater thermal instability and more potential energy for deep convection (i.e. warm, humid air lying below cold, dry air). There are many other measures of convective instability that could be used, but the θe difference provides a first-order estimate of potential instability. Note that I used the 3pm AKST Fairbanks sounding data to get the 500 mb conditions, and I restricted myself to thunderstorms observed between 9am and 9pm AKST.
The chart below shows how the frequency of thunderstorms is related to the daily (9am-9pm) maximum in θe difference. As we would expect, thunder is more frequently observed when the atmosphere is more unstable, and thunder is very rare when there's no instability (negative θe difference).
Now we can address the question of whether there is a notable early June increase in instability that might explain the arrival of thunderstorms; the chart below shows the frequency of moderate (≥6°C θe difference) instability by date. The answer appears to be a resounding "no", as substantial instability is surprisingly infrequent in early June and doesn't peak until a month later. It seems that rising instability is not an adequate explanation for the climatological surge of thunderstorm activity in Fairbanks; and so the search for a suitable explanation will have to continue.
One redeeming aspect of the chart is that the gradually decreasing instability in July and August does mirror quite nicely the drop-off in thunderstorm frequency in late summer. So it seems we can explain the end of the season in terms of the elimination of instability; it's the beginning of the season that is a bit of a mystery.