Tear oyster mushrooms into strips. Mix miso, soy sauce, mirin, olive oil, and sugar in a medium bowl until well combined. Pour sauce over mushrooms and toss together, making sure mushrooms are fully coated. Spread mushrooms on a sheet pan in one even layer. Roast for 15-20 minutes or until caramelized, tossing mushrooms halfway through.
Meanwhile, make the cucumber pepper relish. Stir together rice vinegar with sugar, stirring until sugar has dissolved. Pour over cucumber and peppers, stirring to combine. Let sit for 10 minutes. Add cilantro and sesame seeds, stirring to combine.
Make miso sesame vinaigrette. Whisk together olive oil, yogurt, rice vinegar, miso, honey, sesame oil, and grated ginger until well combined.
Assemble sandwiches. Top half of the bread slices with fresh arugula and roasted mushrooms. Drizzle a generous amount of miso sesame vinaigrette on top of mushrooms and pile on cucumber relish and pickled cucumbers. Drizzle extra miso sesame vinaigrette on top followed by the remaining bread. Serve immediately.