Thoroughly mix ground turkey with brown sugar, a good thwack of salt, along with the ginger and fish sauce. Let it sit while you chop all of your herbs and avocado, then arrange lettuce on a large serving plate.
Heat the oil in a large skillet over high heat, press the meat into the skillet to make a large, thin hamburger. Cook until the bottom has a nice brown crust, about 5 minutes, then flip and cook until the meat is cooked through, about another 4 minutes. Remove from the skillet and let cool for 5 minutes before slicing.
Scatter turkey over the lettuce, top with herbs and avocado, then drizzle with lime juice and serve.