After going through the following links, I have found some interesting numbers (not more than others) about how fast was Hamilton when came alongside Vettel.
https://www.youtube.com/watch?v=j0e2qGVIuL0
https://www.youtube.com/watch?v=7GsDk1B4esE
Lap 31
Vettel - 23.788 - 32.070 - 29.415 (15 laps old Softs)
Hamilton - 23.687 - 32.112 - 29.724 (10 laps old mediums)
Lap 32
Vettel - 23.699 - 31.821 - 29.673
Hamilton - 23.685 - 33.118 - 29.751 (Seems like Hamilton made a mistake in S2 and lost a second)
Lap 33
Vettel - 23.711 - 31.910 - 29.869
Hamilton - 23.662 - 32.182 - 29.724
Lap 34
Vettel - 23.739 - 32.138 - 30.xxx (estimated S3 time based on previous laps - Vettel started hitting traffic at the beginning of Sector 3 and VSC came out - Gap to hamilton was 7.2 seconds)
Hamilton - 23.505 - 32.013 - 29.7xx (estimated S3 time based on previous laps)
On lap 35, Vettel did 23.739 - 31.910 - 29.869 in S1-S2-S3.
On lap 37, when VSC ended, Vettel did 26.402 - 32.084 - 33.057 in S1-S2-S3.
-Vettel's times in S2, before and after VSC are almost similar. He lost 2.6 seconds in S1 for his in lap. He was in the middle of straight when VSC ended. So, his start for S1 was slower. That explains that 2.6 seconds loss.
-Vettel's time loss of 3.2 seconds in sector 3 is due to the fact that, he was entering pits and the sector 3 ends after a driver enters the pits. So he did not made any mistakes, just that he was trundling in pits.
-While Vettel was going slower in S3 on his in lap, Hamilton was going full speed for his out lap, which was not the case while Hamilton was going in pits. Vettel was under VSC. The VSC ended while Hamilton was entering the spot where PIT limit speed applies.
-Vettel's loss is, 2.6 seconds where he started S1. If the VSC would have ended when Vettel was exiting last chicane (instead of middle of straight), there would have been no time loss.
-So Mercedes calling Hamilton at the very last moment, gave them 2.6 seconds advantage. Their pit stop was 0.5 seconds faster than that of Vettel.
-Before pitting, Hamilton was trailing by 7.9 seconds and after Vettel pitted and came out, they were side by side. So, hamilton covered 5.3 seconds on his out lap up to turn 1 where he met Vettel (two thirds of S1, full of S2 and S3 of lap 37 and one third of S1 on lap 38).
There was no magic 6 seconds on one lap, but a combination of many.
-When Hamilton came out of the pits, the gap was 23.909 seconds and Vettel was approaching turn 5.
Read following if you are interested in knowing how the 1.21.xxx would have been possible. Otherwise, ignore.
Lap 34 (Extrapolation of S3 and the lap time - 1:25.218)
Hamilton - 23.505 - 32.013 - 29.7xx (estimated S3 time based on previous laps)
Lap 37 (Extrapolation of S1 and the lap time - 1:21.959)
Hamilton - 22.800 - 30.792 - 28.367
Compared to lap 34, he was faster by 1.7 seconds in S2 and was faster by 1.3 seconds (extrapolation used for S1), I am guessing he would have been faster by 0.7 seconds on S1 (shorter distance and faster part compared to S2 and S3.
He did 22.712 on the lap where he overtook Vettel, with DRS and on next lap he did 23.313, so on his hot out lap, he could have done a 22.800).