EMB-505 Captain reported encountering wake turbulence from a preceding B737 at FL345 that resulted in an upset event.

Narrative

Wake turbulence encountered at 34;500 about one and a half minutes behind a Boeing 737. We were slowed by ATC and eventually descended to the same altitude of the 737. The winds aloft were approximately 30 degrees quartering headwind left to right. Our course was 360 and winds aloft were 330 at 16 kt. We were given a descent and at 34;500 and 290 kt. indicated we felt a small bump; followed by a 5- to 10-degree uncommanded bank to the right immediately followed by a bank of more than 45 but less than 60 degrees right. The pilot flying disconnected the autopilot and rolled the aircraft back to the left without further incident. The pilot monitoring alerted ATC to the wake encounter and requested a turn to the left. ATC hesitantly complied and alerted us to other traffic near our course. ATC indicated that we would be following the 737 all the way into PIT and requested us to return to our course. We asked for a 2-mile offset parallel course to our filed course. ATC did not seem to understand our request and simply said 2 more miles left then direct to a fix further down the line." ATC seemed unconcerned with our report of the wake turbulence. We did not have any other encounters with wake and remained vigilant of the aircraft in front of us."