Found with ITA’s month-long search (see new user thread if you’re not familiar with ITA). Select dates outbound July, stay about 1 month. Have not verified beyond ITA yet.

$826 Dulles-Heathrow booked from, BMI (operated by United Airlines)
$773 Boston-Belfast, Continental
$797 Chicago O’Hare-Milan, Air One
$796 LaGuardia-Heathrow, Air Canada
$686 Newark-Paris, Air India
$759 Newark-Belfast, Continental
$780 JFK-Shannon, Delta
$795 NYC-FRA, United (offers from both Newark and JFK)
$940 MIA-DUS, LTU

(Have not yet checked MEM or gateways west of MSP or DEN).