Place mushroom caps gill side up on a large plate. In a small bowl, whisk together olive oil,vinegar and garlic. Using a pastry or silicone brush,smooth olive oil mixture evenly over mushrooms; let mushrooms marinate, covered at room temp for 1 hour.
Meanwhile, heat canola oil in a large skillet set over medium heat. Add onion when oil starts to shimmer and cook for about 25 minutes, stirring frequently, so they soften and brown evenly. When onions are close to being done, push them aside in the pan and add red peppers. Cook until peppers start to soften slightly, about 7 to 10 minutes. Stir in salt and thyme; cook until thyme is fragrant, about 30 seconds more. Remove from heat.
Set a grill pan over medium-high heat. Place mushrooms gill side down on pan, and cook for about 2 to 3 minutes. Turn mushroom caps gill side up and cook for another 2 to 3 minutes. The mushrooms will be cooked through when they start to soften and liquid starts to appear in the gills. Place them on a cutting board to cool slightly; then cut into strips.
Lie flatbreads out on a cutting board and fill each with sliced mushrooms and a 1/4 of the caramelized onion-pepper topping. Roll the flatbread up, and cut in 1/2 to serve.