In a bowl, toss the thighs with the mustard, vinegar and 1 tbsp. of the EVOO to coat.
Heat a medium skillet over high. Add the remaining 2 tbsp. EVOO and the mushrooms and cook, stirring occasionally, until mushrooms are browned, about 5 minutes. Season with salt and pepper, and cover to keep warm.
Heat a grill or grill pan to medium. Grill the buns, cut sides down, until toasted and golden, 1 to 2 minutes. Grill the thighs, covered, until grill marks develop, about 5 minutes; flip and grill 3 minutes longer. Meanwhile, grill Canadian bacon, 2 minutes per side. Place a half slice of cheese on each thigh and on each piece of bacon; cover and cook until cheese melts, 1 to 2 minutes.
Divide the mushroom mixture among bun bottoms, then stack 2 thighs, 2 pieces of Canadian bacon and 1 tomato slice on each. Add the watercress and bun tops.