Adjust oven rack to middle position and heat oven to 350 degrees. Whisk eggs, milk, Parmesan, 1 tablespoon oil, tarragon, and ¼ teaspoon salt in bowl until well combined.
Heat remaining 1 tablespoon oil in 12-inch oven-safe nonstick skillet over medium-high heat until shimmering. Add broccoli, shallot, turmeric, ¼ teaspoon salt, and ¼ teaspoon pepper and cook, stirring frequently, until broccoli is crisp-tender and spotty brown, 7 to 9 minutes. Stir in water and lemon zest and juice and continue to cook, stirring constantly, until broccoli is just tender and no water remains in skillet, about 1 minute longer.
Add egg mixture and cook, using rubber spatula to stir and scrape bottom of skillet until large curds form and spatula leaves trail through eggs but eggs are still very wet, about 30 seconds. Smooth curds into even layer and cook, without stirring, for 30 seconds. Transfer skillet to oven and bake until frittata is slightly puffy and surface bounces back when lightly pressed, 5 to 8 minutes. Using rubber spatula, loosen frittata from skillet and transfer to cutting board. Let sit for 5 minutes before slicing and serving.