How can I interpolate large segments of missing data?

Unfortunately, there's not much to be done. Sea ice varies so much from year to year, that interpolating from surrounding years may not give a
representative answer. The best thing to do would be to simply leave the missing data out and if you're doing a timeseries analysis, do a type
that can handle missing data.

If you need to fill in the gap, it depends on what is being looked at. If it is total extent or area values (i.e., single daily values summing
up all the ice over each hemisphere), then just drawing a straight line between missing data points is probably not a bad approximation.

If you need to actually look at daily fields, then it's more difficult. The best thing to do might be to fill in with National Ice Center weekly
ice charts. Three issues with these: (1) They're only weekly, so you'd still have to interpolate to get daily values, but it's easier to
interpolate through one week. (2) They are based off of different data, so they are not consistent with the passive microwave data. Thus you'd
need to compare surrounding periods to find the offset between them and make appropriate adjustments. (3) They're on EASE-grid, not polar stereographic grid, so you'd have to regrid one or the other to compare.

The National Ice Center charts referred to can be acquired from the National Ice Center website.