Heat the oil in a small nonstick skillet over medium heat. Divide the sausage in half and form each piece into a thin 4-inch patty. Add the patties to the skillet and cook until golden and cooked through, about 2 minutes per side. Transfer to a paper towel-lined plate. Reserve the skillet and drippings.
Whisk together the eggs with the cream cheese, chives, cilantro, parsley, a sprinkle of salt and a few grinds of pepper.
Heat the skillet and drippings over medium heat. Add the eggs and stir for the first 30 seconds with a silicone spatula to break up the curds as they cook. Stop stirring and let the eggs cook another 30 seconds (they will still be wet on top), then slide the omelet onto a plate. Flip the eggs back into the skillet so the uncooked side is face down. Top one side of the eggs with the Cheddar. Cook until the eggs are cooked through but still soft, another 30 seconds to 1 minute.
Put a sausage patty "bun" on a plate. Using the silicone spatula, fold the omelet in half, then into quarters before sliding it out of the skillet and onto the sausage patty. Top with the sliced avocado and the second sausage patty "bun."
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