Whisk the eggs together with the milk and salt. Set to one side as you melt the butter in a medium saucepan over a medium heat.
Meanwhile, hollow out the brioche buns, making a cavity where the scrambled eggs will sit. Add the whisked egg mixture to the saucepan and cook over a medium heat, stirring occasionally until the egg is set.
Meanwhile, dress the smoked salmon strips with the lemon juice. Spoon the scrambled egg into the cavity of the brioche buns, then top with a some chopped chives and strips of the smoked salmon. Season the salmon with black pepper, then arrange the brioche on serving plates.
Rest the reserved tops against the buns and finish by garnishing the brioche with a chive stalk. Serve immediately.