Preheat the oven to 350°F. Line a 9x13-inch light-colored metal baking pan with parchment paper, leaving enough overhang on the sides to lift the bars out later.
Brown the butter. Add the butter to a light-colored saucepan and cook over medium heat, stirring or swirling often. The butter will melt, foam, and eventually develop golden-brown milk solids on the bottom and a nutty aroma. As soon as it is deeply golden, remove it from the heat and pour it into a heatproof bowl, scraping in all those browned bits. Let cool slightly.
Make the crust and crumble: In a large bowl, whisk together the flour, granulated sugar, brown sugar, baking powder, salt, and cinnamon. Make a well in the center and add the browned butter, egg, and vanilla. Mix until a crumbly dough forms.
Reserve the crumble: Set aside about 2 cups of the dough for the topping. Press the remaining dough evenly into the bottom of the prepared pan.
Par-bake the crust: Bake for about 15 minutes, or until the crust looks set and is just beginning to turn lightly golden around the edges. Remove from the oven and set aside.
Make the apple filling: Add the diced apples, honey, brown sugar, lemon juice, vanilla, cinnamon, nutmeg, salt and cornstarch to a large skillet or saucepan. Cook over medium heat, stirring occasionally, until the apples begin to soften and release their juices but still hold their shape and the juices thicken and become glossy. Remove from the heat.
Assemble the bars: Spoon the apple filling evenly over the par-baked crust. Scatter the reserved crumble over the top, squeezing some of it together with your fingers to create a few larger crumbles.
Bake again: Return the pan to the oven and bake for 30 to 40 minutes, or until the crumble is golden brown and the apple filling is bubbling around the edges. Start checking at 28 minutes if it's your first time making these.
Cool completely: Let the bars cool in the pan for at least 2 hours. Use the parchment overhang to lift them out of the pan, then slice into bars.
Serve: Dust with powdered sugar, if desired, and serve at room temperature or slightly warm.