EMB-505 Captain reported the flight crew failed to maintain a constant climb gradient during departure; causing ATC to alert the flight crew to terrain and climb immediately. The flight crew executed the procedure and continued the flight.

Narrative

Departing ZZZ on Runway XX on ZZZZZ Departure. Prior to engine start a full performance review was done and a full crew briefing to follow. We determined the need for a special engine out obstacle departure procedure in the event of an engine failure after takeoff. The all-engine operating departure plan as well as the engine failure departure plan were briefed prior to engine start. The departure clearance was climb via the ZZZZZ with a top altitude of 15;000 ft. per the departure procedure. During the crew brief and cockpit setup; 15;000 ft. was set in the altitude selector. With our takeoff clearance we were told; Cleared for takeoff Runway XX climbing through 10;000 ft. contact Departure."At some point during takeoff roll the altitude in the altitude selector was changed by the pilot monitoring to 10;000 ft. Shortly after takeoff Tower told us to contact Departure. Checking in with Departure; the pilot monitoring checked in saying; "Aircraft X climbing 10;000 ft.;" to which I believe we heard something along the lines of higher in a moment. Once I realized the altitude had been changed; I thought it was odd but with no correction from the controller after check on and me believing I heard that we were to expect higher soon; I pressed on - but decided to use extreme caution by closely monitoring the terrain awareness system in the airplane as well as using the visual conditions outside to maintain separation with the terrain and use that display to get further away from the terrain but still remain on the route for the SID. Approaching ZZZZZ1; the Supervisor came over the radio and advised us of a low altitude alert and told us to climb to 15;000 ft. immediately. He then told us that we were required to maintain a climb gradient for that departure. We knew this and knew that we had the performance to comply with the required climb gradient and advised him that we were given 10;000 ft. after departure. The TAWS system in the aircraft never alerted during the event."