Preheat the oven to 350°F (177°C). Line a 9×13-inch baking pan with parchment paper, leaving an overhang on two sides so the finished bars can be lifted out easily.
In a medium bowl, stir together the melted butter, granulated sugar, vanilla, and 1/2 teaspoon salt until smooth. Add 1 3/4 cups flour and mix just until no dry flour remains. Press the dough evenly over the bottom of the prepared pan. Bake for 15 minutes, until the surface is set and the edges are just beginning to turn pale golden. Keep the oven at 350°F.
While the crust bakes, place the sliced apples in a large bowl and toss them with the lemon juice. Add the light brown sugar, cornstarch, 1 1/2 teaspoons cinnamon, nutmeg, and 1/8 teaspoon salt. Toss until the apple slices are evenly coated. Prepare this mixture shortly before assembly so the apples do not release excessive liquid while standing.
In another bowl, combine 3/4 cup flour, rolled oats, light brown sugar, 1/2 teaspoon cinnamon, and 1/4 teaspoon salt. Add the cold cubed butter. Work the butter into the dry ingredients with a fork, pastry cutter, or fingertips until moist crumbs form with a few pea-size butter pieces remaining.
Spread the apple mixture evenly over the warm prebaked crust, arranging the slices in a fairly even layer. Sprinkle the oat streusel loosely over the apples without pressing it down. Bake at 350°F for 35 to 40 minutes, until the streusel is golden, the apple juices are lightly bubbling around the edges, and a thin knife inserted into the center passes through tender apple slices with little resistance.
Place the pan on a wire rack and cool at room temperature for 1 hour. Refrigerate the pan for another 1 hour so the layers can set. Lift the bars from the pan using the parchment overhang and cut into 16 equal bars.