In a medium saucepan over medium heat, melt the butter and then continue cooking until it begins to turn golden brown and smells nutty, about 7 minutes, stirring frequently to prevent burning.
Move the brown butter pan off the heat and immediately whisk in the sugar, brown sugar, and salt. Let the warm butter-sugar mixture cool for another 10 minutes, so it doesn't scramble the eggs. Once cooled to room temperature, whisk the eggs one at a time into the browned butter-sugar mixture, then add the vanilla.
Using the whisk or a spatula, stir the oats, coconut, and chocolate/fruit/candy into the filling mixture. Pour the filling over the parbaked crust, ensuring it is evenly spread over the top. It will look relatively thin.
Bake the bars for 15-20 minutes (start checking at 15 but they will probably take 20). The bars are done when the top is firm and golden and the middle barely jiggles/is set when the pan is shimmied.
Let the bars rest at least 15 minutes before cutting and serving. They are much easier to cut when cold, but they are (obviously) at their gooiest when warm.
Dust the bars with a sprinkling of powdered sugar (if desired) and serve. Baked gooey oat bars will keep for 2 days tightly wrapped at room temperature or 4-5 days covered in the refrigerator.