Add the eggs, milk, tarragon, salt, and pepper to a medium bowl. Beat together with a whisk or a mixer on low speed until combined. Set aside.
Melt butter in a cast iron skillet over medium heat.
Add lobster, chives, and a pinch of salt, stirring occasionally until mixture is heated through, or about 3 minutes.
Turn the heat to low and add the eggs. Scramble them by using a spatula to occasionally pull cooked eggs from the pan until the eggs form curds. The mixture should take about 5 minutes to cook; eggs will be done when they’re somewhat firm and no longer runny.
Garnish with a few more chives, if desired. Serve immediately.