Cut mushroom stems crosswise into 1-inch pieces--you should have at least 16 pieces total. (Reserve the caps for another use.) Using a sharp paring knife, score both flat sides in a cross-hatch pattern.

Whisk broth, wine, seaweed, miso, tamari (or soy sauce), maple syrup and Worcestershire in a small saucepan. Heat over medium heat, whisking occasionally, until the miso dissolves, 1 to 2 minutes. Add the mushrooms and reduce heat to maintain a bare simmer. Cook, stirring occasionally, for 30 minutes. Remove from heat and let cool, stirring occasionally, for 1 hour.

Using a slotted spoon, transfer the mushrooms to a paper-towel-lined plate and pat dry. Sprinkle with salt.

Heat oil in a medium nonstick skillet over medium-high heat. Add the mushrooms and cook, turning once, until browned on both sides, about 3 minutes total. Serve the mushroom "scallops" with the sauce, if desired.