Air carrier Captain reported they temporarily lost control of the aircraft while initiating an unexpected go around.

Narrative

Conducting ILS RWY XXR in ZZZ. At approximately 1000'; Tower instructed us to terminate the approach; maintain rwy heading; and climb to 3000 feet. Upon pushing the TOGA button; and announcing GO-AROUND; FLAPS15; CHECK THRUST; I did not adequately push to control pitch attitude. Pitch increased rapidly to 15 degrees nose up; causing airspeed decay to the top of the yellow band. While correcting my self induced PIO (Pilot-Induced Oscillations); I lost approximately 250-350 feet while conducting a climbing right turn eventually to 090 degrees. We were then given a climb to 5000 feet; and conducted a second approach without incident."