Hmm... have you considered taking your 1st level in fighter and then switching to monk? You'd have delayed extra attack and miss out on level 12 ASI but no actual monk features. As a benefit you'd have heavy armor, shield and all martial weapon proficiencies upfront, which would free you up to round up your Strength to 18 with Heavy Armor Master at 4th level. Also you'd get dialogue options for two classes.
That was my first thought, but then I realised that heavy armor works kinda bad with multiclassing into fighter/monk, because you need 13 DEX and 13 WIS in order to make this transition. And DEX is much less useful to the heavily armored character. I am leaning towards Warlocke's suggestions about focusing on the defense and compensating for the damage by wielding a two-handed greatsword which is much better than monk weapons, 2d6 against 1d8. Then I'll probably branch into a shadow monk to get darkvision which githyanki don't have and to get shadow step for consistent mobility (I won't use my BA for flurry of blows anyway). Having the disadvantage on stealth rolls bothers me a bit as a shadow monk though Larian could introduce some sort of a mithral armor. We already have the adamantine splint mail that is heavy and has no stealth penalty!
