To make the sandwiches, preheat the oven to 350°F. To shred the mushrooms, run a fork from the top of a mushroom stem to the bottom, repeating until you have shreds that resemble pulled pork. Pull shreds into separate pieces. Roughly chop mushroom caps. Arrange all mushroom pieces on a rimmed baking sheet. Toss with oil and a pinch of salt and pepper, using hands to coat mushrooms thoroughly. Roast until mushrooms start to dry out and brown in places, 25 to 35 minutes.
Put broth and onions in a pan over medium-high heat, and cook, stirring occasionally, until onion starts to turn translucent, about 2 minutes. Add a little more broth if needed to prevent onions from sticking to the pan.
Add shredded mushrooms to the pan and cook 2 minutes more, stirring occasionally. Add small amounts of broth if needed to keep mushrooms from sticking. They should be moist but not boiling in liquid. Stir in vinegar, liquid smoke, thyme, garlic granules, chili powder, salt and pepper. Cook 2 minutes more.
Lower heat and stir in barbecue sauce. Remove from heat, cover to keep warm and set aside while preparing the coleslaw.
To make the avocado coleslaw, mash avocado with mayonnaise until smooth in a large bowl. Stir in agave nectar, vinegar, lime juice, salt and pepper. Add coleslaw mix and toss to combine well.
To serve, spoon mushroom mixture onto buns and serve with coleslaw on the side.