Mountbatten didn't have two ships sunk under him, he had one sunk (and heavily damaged) and one heavily damaged.
On 9-10 May HMS Kelly was struck by torpedoes amidships from the S-boat S-31. Mountbatten transferred his flag to HMS Javelin while repairs were undertaken. Even D K Brown mentions Mountbatten's quick thinking (and knowledge of the ship going back to when she was designed) which saved Kelly in Nelson to Vanguard. Mountbatten had a pre-arranged list of topweight that he knew would be needed to jettison to prevent her capsizing, he did this and with luck the ship was saved.
On 29 November 1940 his Flotilla engaged three German destroyers off Lizard Point. Mountbatten turned to port to match a German course change, this disrupted the directors and they lost their target, the result being two torpedoes hitting the ship as well as enemy gunfire, losing both the bow and stern sections (he lost 200ft of his ship - very careless) but again by exercising good damage control was towed back to harbour and rebuilt. Of 15 RN destroyers hit by two torpedoes during the war, only two survived, one of those being Javelin.
The following month Kelly was now repaired so he went back aboard her. Kelly was sunk by dive bombers on 23 May 1941 off Crete and capsized during a high-speed turn but they did shoot down three Ju 87s and damaged another, maybe not a fair return but not a bad effort given the AA armament and directors then in use. Of course the film In Which We Serve captures this event.
There is no doubt that that he was a very much a 'press-on' type of Commander, he certainly wasn't timid. Whether either events while performing turns were poor judgement or just bad luck mixed with effects of battle damage is open to interpretation. What is more certain is that he had the knowledge and skill to keep both Kelly (in 1940) and Javelin afloat following appalling damage, arguably saving the loss of two ships early in the war.