Preheat the oven to 450°F. Line a sheet pan with foil. Wash and dry the fresh produce. Halve the potatoes lengthwise, then cut crosswise into 1/2-inch pieces. Transfer to the foil. Drizzle with olive oil and season with salt and pepper; toss to coat. Arrange in an even layer. Roast 18 to 22 minutes, or until browned and tender when pierced with a fork. Remove from the oven.

Place the mushrooms in a bowl; cover with hot water. Set aside to rehydrate at least 10 minutes. In a bowl, whisk together the miso paste and mayonnaise. Taste, then season with salt and pepper if desired. Thinly slice the cucumber(s); place in a separate bowl. Add the vinegar and season with salt and pepper. Set aside to marinate, stirring occasionally, at least 10 minutes; taste, then season with salt and pepper if desired.    

Drain the rehydrated mushrooms. Transfer to a cutting board, then finely chop. In a bowl, combine the beef and chopped mushrooms; season with salt and pepper. Gently mix to combine. Form the mixture into 2 patties (or 4 patties for 4 servings), about ½-inch thick. In a medium nonstick pan (or large for 4 servings), heat a drizzle of olive oil on medium-high until hot. Add the patties. Cook 4 to 5 minutes per side (flipping carefully, as the oil may splatter), or until browned and cooked through. Leaving any browned bits (or fond) in the pan, transfer to a work surface.

Working in batches if necessary, add the halved buns, cut side down, to the pan of reserved fond (if the pan seems dry, add a drizzle of olive oil). Toast on medium-high 30 seconds to 1 minute, or until lightly browned. Transfer to a work surface. Assemble the burgers using the toasted buns, miso mayo, cooked patties, hoisin sauce, and marinated cucumber(s). Serve the burgers with the roasted potatoes on the side. Enjoy!