In a large skillet, heat the olive oil over medium heat until it begins to shimmer.
Add mushrooms to the pan and cook, stirring occasionally until soft and deeper in colour, approximately 5 minutes.
Add the miso paste, the sherry vinegar and a splash of water (about 2 tablespoons) to the pan and stir well, allowing the miso paste to dissolve into the liquid and coat the mushrooms.
Lower the heat to medium-low and simmer gently, completely cooking the mushrooms and reducing the miso liquid to a glaze. Allow to cook for an additional 4-5 minutes.
Add the fresh thyme and pepper to the mushroom mixture and remove from heat. Allow to rest while you prepare the rest of the sandwich.
Gently toast the slices of Stonemill Bakehouse Sourdough Multigrain Rye bread until lightly golden brown.
Spread a thick layer of the white bean dip on each slice (approximately 2 tablespoons) and top with ¼ of the warm mushroom mixture. Repeat with remaining slices.
Top each tartine with a drizzle of olive oil and serve.