ERJ-175 Captain reported an upset occurred climbing through FL356 when they encountered wake turbulence from a preceding B777 at FL360.

Narrative

During climb to filed altitude; we were held at FL340 for an extended period of time due to a heavy Boeing 777 overtaking us at FL360. This was not initially communicated by ATC; so we asked when we could expect our final filed altitude of 360; and this is when they informed us of the heavy [aircraft] above. As the heavy passed; we saw that it was directly above us on the same lateral track; so we patiently waited to go higher. The aircraft moved to somewhere between eight and ten miles ahead of us when ATC cleared us to climb to FL360. I set FL360 in our altitude selector and initiated a Flight Level Change as my FO (First Officer) and I both confirmed the altitude. The climb was slow and steady; we were heavy with a lot of fuel for a long flight. As we started passing through about FL356; the plane rolled about 5 degrees to the left; then rolled slowly to the right; past the 30 degree roll angle as we were buffeted with wake turbulence. I called UPSET and took the controls; disengaging the autopilot and stabilized the roll; allowing the nose to come down through the horizon and began a descent away from the turbulence; asking my FO to inform ATC we were descending out of the wake and returning to FL340. Discussing with ATC; we asked them for an offset course; and elected to stay at FL340 for the duration of the flight. Once the aircraft was in a stable state and the automation was re-engaged; my FO checked in with the cabin crew who informed us that the only ones who had been up and out of seats were the flight attendants. They immediately sat down when it was apparent something was wrong; and no one was injured. The flight continued from there unhindered. The cause seems to have been wake turbulence from the preceding heavy 777.Suggestions: I've been under a heavy aircraft while being overtaken by it in the past a few times. When we were cleared to climb; the separation looked like it was big enough for us to avoid any serious turbulence from the heavy based on previous experience. Looking back; I should have and will moving forward ask to be offset from under aircraft of this size; preferably upwind; so as to give a greater margin for avoiding this kind of turbulence if it lingers in the air longer than we expect.