I had a half-completed build for a Captain America style brawler. Tavern Brawler for the d4 unarmed and proficiency with Improvised Weapons, Dual Wielder for the ability to TWF even if the weapons aren't light--and then use TWF with fist and shield.
Unfortunately, by a strict reading of the rules it doesn't work. A shield is an object. And while it can be used as an improvised weapon, it is certainly not a "one-handed melee weapon"--which is the requirement for TWF.
Still: a Tavern Brawler can definitely bash with that shield at full proficiency bonus for (at least) d4 damage. D6 if you and your DM agree it should be treated more like a mace than a club (iron-banded shield).
So go with human, fighter 1 (Duelist for the +2 damage), Rogue 1 or Bard 3 (for Expertise in Athletics) and/or Barbarian 1 for Rage (advantage on Str checks), Tavern Brawler and Shield Master, and then:
1. Grapple them with free hand (attack action).
2. Knock them down (bonus action shove).
3. In subsequent turns beat them with the shield, or kick with Unarmed Strike for at least 1d4 + 2 + Str.
Really, at that point just go ahead and follow the excellent advice here:
http://community.wizards.com/forum/player-help/threads/4142801
Oh, also: up above someone asked if you can grapple with your shield hand. No. Grappling requires "at least one free hand."