Toss the radishes, tartar sauce, sliced celery and all but a few celery leaves in a medium bowl. Season with salt and pepper and set aside.
Whisk the eggs in a large bowl with 1/4 cup water, 1/2 teaspoon salt, and pepper to taste. Heat 1 tablespoon butter in a large nonstick skillet over medium heat. Add the scallions and cook until slightly wilted, about 1 minute. Add the egg mixture and cook, stirring, until just beginning to set, about 3 minutes. Add the herbs and cheese and continue cooking until the cheese melts, about 2 more minutes. Transfer to a bowl and cover to keep warm.
Wipe out the skillet and return to medium-high heat. Spread the outsides of the buns with the remaining 1 tablespoon butter; add to the skillet and toast until golden, about 1 minute per side. Fill each bun with a lettuce leaf and some of the scrambled eggs. Top with the remaining celery leaves. Serve with the celery salad.