Monk, QS and fist.
Base is 1d8+1d4+2dex
Ki is worth 1d4+dex; get 1 Ki/short rest/level
Level 5 is 1d8+dex and +1d6-1d4 on both MA and Flurry.
Fighter, dex, TWF, BM.
Base is 2d6+2dex
Level 2 is +1d6+dex/SR
Level 3 is +4d8/rest
Level 5 is +1d6+dex/turn + 1d6+dex/rest
So at 5 we get with 18 dex
3d6+3dex (22.5)/turn + 2d6+2dex+4d8 (33)/rest on BM
on monk:
2d8+1d6+3dex (24.5)/turn + 5d6+5dex (37.5)/rest on Monk (no subclass)
At 3 w 16 dex
2d6+2dex (13)/turn + 1d6+dex+4d8 (24.5)/rest on BM
1d8+1d4+2dex (13)/turn + 3d4+3dex (16.5)/rest on Monk
Now, monk Open Hand abilities aren't bad (I'd say a touch better status effects than BM; they can force 6 dex-or-prone saves/SR, vs BM has 4), but the gap isn't as large as above.
One idea is to give d6 dice, upgrading to d8s at level 5. That reduces the damage spike a little bit.