Crack the turkey eggs into a large bowl.
Add milk or cream, salt, and black pepper to the bowl and whisk vigorously until fully combined and frothy.
Heat a non-stick skillet over medium-low heat.
Add butter to the skillet and let it melt until it shimmers and begins to foam, then swirl to coat the pan.
Pour the egg mixture into the hot skillet and let it sit undisturbed for about 30 seconds, until the edges begin to set.
Using a rubber spatula, gently push the cooked edges toward the center and tilt the pan to allow uncooked eggs to flow underneath; continue folding and pushing until eggs are mostly set but still moist and glossy, about 3–5 minutes.
Remove the pan from heat (eggs will continue cooking off the heat) and transfer the eggs to serving plates.
Garnish with fresh chives or parsley if desired, and serve immediately.