Monks in the D&D sense of the mystical martial artist can be a bit indirect to replicate. I think that the best playbook to start from would be Warrior, since martial mastery is a significant goal here.
From there, if everyone at the table is in agreement that in your world people can train their bodies to the point to be able to achieve supernatural effects... then you can just go for it! Use the write-in spaces under Take the Risk or Mettle to make special abilities that you think a monk would be able to do. You can take inspiration from other playbooks for some of these, like Scoundrel's "to dodge or evade harm" Take the Risk option or the Wildsoul's "move as swiftly as the wind" Wisdom ability.
You can do similar things with equipment. Warriors inflict a lot of harm with "a sturdy weapon" so you can easily go for the brass knuckles or other iconic monk weapons (like a bo staff, using the Wildsoul's staff as an example)... but if everyone at the table is on board with it, you could have your weapon by "Years of hand-to-hand combat training" with maybe the forceful or grasping tags. (This would overrule Warriors typically using the median die when inflicting harm with their body, because your whole body has become a sturdy weapon!) Similarly, you might have "Unencumbered evasion (1 Armor)" in place of the Warrior's usual armor option, showing that you're intentionally traveling lighter (while still accounting for it by "using up" a load checkbox).
As your character progresses, something like a monk would be a good opportunity to make use of an archetype. You could gain more magical powers through Forsworn (and maybe your oaths are to your monastic order, rather than a deity or demon), while something like Martyr works well for a character that's trained their body to its absolute limit and beyond.
Let me know what you think, and I really hope you have a great time playing Fast Fantasy!