Place butter pieces in a medium skillet. Melt the butter over medium heat, swirling it in the pan occasionally. The butter will foam and pop, so be careful. Continue to swirl the pan often. Remove pan from heat once the butter starts to brown and smells nutty. There will be small brown bits on the bottom. The butter should be an amber color. Pour butter into a medium bowl.
Add the vanilla extract and brown sugar to the brown butter. Stir until smooth. Add the flour, oats, pecans, cinnamon, nutmeg, and salt. Stir until combined. Set aside.
Cut the apples in half from stem to end. Use a melon baller or small spoon to scoop out the apple core and seeds. Dig deep enough so there is a little room to "stuff" the apples with the filling. Remove the apple stems.
Pour the hot water in a large cast iron skillet or baking dish. Arrange the apples in the baking dish with the flesh facing up. Sprinkle brown sugar and cinnamon evenly over the apples. Divide the streusel topping between the apples, pressing down so it doesn't fall out.
Cover the pan with aluminum foil and bake the apples for 20 minutes. Remove from the oven and carefully remove the foil. Bake for 30-40 more minutes or until the apples are soft and the topping is golden brown. You want the apples to be soft, but not mushy.
Remove from the oven and serve warm with vanilla ice cream and salted caramel sauce, if desired.