A solar eclipse is scheduled to pass over the Faroe Islands and this seems like an excellent chance to visit.
What is a cheap way of flying there from mainland Europe? Kayak seems to only find ridiculous $2,000 fares to Sorvagur airport.
The only airline with regular scheduled flights to the Faroe Islands is Atlantic Airways, the flag carrier. There are daily flights to/from Copenhagen and less frequently to/from Keflavik, Bergen and Billund.
However, they are probably already fully booked for the days near the eclipse ...
Your best bet may be to find a dedicated eclipse tour with a chartered flight.