In a small saucepan over high heat, bring 1/4 cup of water to a boil. Add the pea shoots, put a lid on the pan and remove it from the stove burner. Allow the pea shoots to wilt for 2 minutes. Drain and discard any remaining water from the pea shoots and reserve.
Add the eggs, milk, salt and pepper to a small dish and whisk to combine.
Heat a 10-inch omelet pan or skillet with slopping sides, over medium-high heat and add 1 tablespoon of oil. Pour the egg mixture into the pool of oil, allowing the egg mixture to spread the oil out.
Cook the eggs until they are set on the bottom and the edges begin to firm up. Using a stiff spatula, lift the edges all around the pan slightly to push the eggs into the center area. When you see that the center is beginning to slightly firm up, about 20-30 seconds, add the pea shoots, tarragon and cheese in a strip, just slightly off-center. Loosen up the edges and carefully, flip the larger side up over the filling. If you allow the eggs to cook too long the omelet will crack at the fold. Reduce the heat to it’s lowest point and continue to cook for another 30 to 45 seconds or until the eggs are firm and no longer runny.
Plate the omelet, garnish with the sour cream and pea shoot sprigs and serve.