Embraer-135 Captain reported a windshear warning during climb resulting in an altitude exceedance in gusty visual conditions. The crew performed the windshear escape maneuver and safely continued the climb out with no injuries.

Narrative

High gusty surface winds all day; northwesterly gusting to 37kts. Moderate to severe on arrival on ZZZZZ. On taxi out surface winds switched to southeasterly; 10 to 15 kts. Taxied out to Runway XXL for departure. Two aircraft arrived as we were awaiting our flow time to ZZZ1. One a Mooney landing on YYR; one a Cessna 340 landing on XXL; with difficulty in gusty conditions. Tower winds were initially from different directions at different locations on the airport. Our flow time came for our take off. Surface winds were now more unison in direction and velocity from the southeast. On take off roll; First Officer (FO) had trouble staying on the centerline due to gusty conditions; but did manage throughout the take off roll. FO held the aircraft on the runway for a higher lift off speed due to the gusty conditions. Once airborne and about 500 ft. we get the WINDSHEAR" warning with moderate to severe turbulence. Together FO and I execute the escape maneuver; autopilot off; full power and pitch high; unable to level off under these conditions at assigned altitude of 8000 ft. we finally get control of the aircraft climbing through 8600 ft. Reported the altitude deviation due to WINDSHEAR; with ZZZ Departure on frequency XXX.X. Moderate turbulence through 14;000 then better above 20;000 ft.Cause: WINDSHEARSuggestions: Waiting even longer after the surface winds showed uniform direction and velocity around the airport. Crew team work mentioned above just to emphasize the quick and correct response from both of us."