I would set up a sort of PLL.

Its output should give a steady sinus without missing periods.
Needs to be locked to the incoming Signal.
Depending on its setup it has same frequency and phase.

Did you try with the FFT.

A spike have to be present in the spectrum.

Using right relationship between time/freqeuncy you can define the frequency and the period.

Looks like a PSK (phase-shift-keying) modulated carrier to me - that leads me to support bholzmayer's suggestion to use a PLL to reconstruct the carrier...

Another thing: Depending on what you mean by "reconstructing the sine wave" the solution might very well be simpler. if you just need to know the frequency of the sine (and not the exact phase) you can just measure the distance of the zero-crossings to get your period of course...

Not quite a PSK case to me because the missing value section is empty

right! OOK then :)