B737NG Captain reported encountering wake turbulence 10 miles in trail of an Antonov-124 aircraft at FL340 that resulted in an upset.

Narrative

Just past the JST VOR at Flight Level 340; our aircraft experienced an aircraft upset. The aircraft first experienced a little bit of turbulence for a second or two then rolled slight to the left. That small roll was immediately followed by an aggressive roll to the left exceeding 40 degrees. The 'Bank Angle' audible warning also sounded. The First Officer quickly executed the Upset Recovery maneuver and the aircraft was returned to level flight losing only about 100 ft. The flight conditions were dusk; VMC on top; smooth air; winds 240 [degrees at] 35 [kts]; flying at Mach 0.75; and the seatbelt sign was still on from some light chop that was experienced a few minutes prior to the event.Once the recovery was complete; I advised ATC of the upset; altitude loss and return to normal flight conditions. I also called the Flight Attendants and was advised of no passenger or crew injuries. One Flight Attendant was in the aisle; but was able to immediately sit in a passenger seat when the aircraft rolled. I talked with the entire crew at the end of the flight; as well as the next morning; to ensure there were not any delayed injuries or other concerns.After talking with the inflight crew and the First Officer after the upset; I queried ATC as to what we were following. They responded that we were following an Antonov 124 large cargo jet. I commented that we most likely experienced a wake turbulence event from that aircraft. They seemed surprised as we were 10 miles in trail on the same route and at the same altitude. After some anecdotal conversation about the size of the Antonov; ATC asked if we wanted additional spacing vertically and laterally the rest of the way in to PHL. We replied affirmatively and were issued a pilot's discretion descent and route offset on the initial portion of the arrival. PHL Approach later vectored the Antonov completely off the published arrival and issued them direct routing to the airport.Moral of the story; an aircraft upset and wake turbulence event can occur at anytime; anywhere; even if you are several miles behind and at the same altitude.