Cut mushroom stems into 1/4-inch rounds then cut the top into strips. Heat the oil in a medium pan over medium-high heat. Add cut mushroom stems to the pan, flat side down, and cook for several minutes without stirring, until lightly browned. Turn mushrooms over and brown the other side. Add mushroom strips, sprinkle with a little salt, then cover tightly and continue to cook for about 4 to 5 minutes longer. Remove from heat and set aside.
In a medium bowl, stir together mayonnaise, lemon juice, Old Bay Seasoning, salt, and black pepper. Taste and adjust seasonings, if necessary. Add the hearts of palm, celery, and herbs to the bowl with the dressing. Transfer the cooked mushrooms directly into the bowl with the hearts of palm and dressing. Gently toss to coat.
To toast the rolls, heat the pan on medium-high heat. Open the rolls and place them face down on the hot pan to toast for several minutes. Remove from heat. To serve, spoon filling into toasted buns. Serve with lemon wedges.