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The very lowest prices seem to be between Geneva and Chicago (starting at just ~ £690.00!), but there are plenty of similar options from Zurich, Basel, Munich, Frankfurt etc to New York and San Juan too (other destinations are probably also possible).
The sale dates are from the end of November to early December, and then from mid February through to mid April.
In order to get the very cheapest prices you need to book via the Italian version of Budgetair, but you can find the deals on other online travel agencies for not much more.
If you would rather book directly, the BA website prices start at about £780.00 for Iberia flights via Madrid.
Flying BA or AA via London Heathrow for ~£20.00 more is probably a better idea as you could choose to ‘miss’ the last segment of your journey, if you don’t check bags (and don’t want the extra Tier Points).
You can also choose to knock another ~ £200.00 off the price in exchange for 35,000 Avios.
Remember that you will also earn Triple Avios at the moment if you book the November-December dates!
There’s still a bit (a lot?) of confusion as to what ‘triple Avios’ actually means, but my current view is that in Premium cabins you essentially end up getting 200% of flown Miles added to whatever the standard earnings would be.
Geneva-Heathrow-Chicago Return in Business Class earns 14,360 Avios, and(I think!) you would get an additional 17,812 = 32,172 Avios in total.
There’s a handy 340 Tier Points to consider too if you’re chasing status!
These are fantastic fares. Having to start outside of the UK is mildly inconvenient, but unless you live in the South East it isn’t really much worse than starting in London anyway.