Crack the eggs into a small bowl. Add the chives, parsley, salt, and pepper to the eggs, then whisk with a fork until frothy.
Heat a 10" non-stick skillet over medium heat. Add the butter, and swirl until it melts then foams. Once the foam subsides, pour the whisked eggs into the pan. Swirl the pan to cover the bottom with an even layer of eggs.
Saute the eggs over medium heat until the edges are firm and the center is just barely cooked through, 3-4 minutes. Deflate any large bubbles with the tines of a fork.
Sprinkle half the parmesan cheese over one half of the omelet. Top the cheese with the lump crab meat, then sprinkle with green onions. Layer the avocado over the green onions, then top with the remaining parmesan cheese.
Using a spatula, fold the empty half of the eggs over to form a half circle. Remove from heat, and transfer to a plate. Sprinkle with additional fresh herbs to taste, and garnish with a dollop of sour cream. Serve immediately.