In a large mixing bowl, combine the celery, mayonnaise, green onion, lemon juice, lemon zest, salt, and pepper. Mix well.
Fold in the lobster, stirring gently just until everything is well combined.
Use a sharp knife to slice the sides of the brioche buns vertically, opening them gently like a book without fully separating the halves. Spread softened butter on the cut sides. Heat a skillet over medium heat and toast the buns cut-side down until golden brown and crisp, about 2-3 minutes.
Fill the toasted buns generously but gently with the lobster mixture to avoid breaking the tender buns.
Garnish with extra sliced green onions, if desired, and serve immediately.