In a medium saucepan, melt 8 tablespoons (1 stick) of the butter over medium heat. Cook, swirling the pan occasionally, until the butter turns golden brown and smells nutty, 4 to 6 minutes. Immediately remove from the heat to stop the cooking. Transfer the butter to a heatproof bowl or measuring cup, scraping in the browned bits from the bottom of the pan. Refrigerate until cooled completely, about 20 minutes.

Meanwhile, melt 3 tablespoons of the butter in a 12-inch cast-iron or ovenproof skillet over medium heat. Add the apple and ½ teaspoon of the apple pie spice and cook, stirring occasionally, until just softened, 5 to 6 minutes. Transfer to a medium bowl and refrigerate until cooled completely, about 15 minutes. Do not wipe out the skillet.

Preheat the oven to 350°F.

In a medium bowl, mix together the flour, baking soda, salt, and ¾ teaspoon apple pie spice. Set aside.

In the bowl of a stand mixer fitted with the paddle attachment, beat the cooled browned butter, brown sugar, granulated sugar, and remaining 8 tablespoons softened butter on medium speed until light and fluffy, about 3 minutes. Add the eggs, egg yolk, and vanilla, beating until smooth and scraping down the bowl as needed.

With the mixer on low speed, gradually add the flour mixture and beat until just combined. Fold in the toffee bits.

Press the dough into an even layer in the skillet. Scatter the apples over the top, pressing them in lightly to adhere. Bake until the edges are golden brown and the center is just set but still slightly soft, 40 to 45 minutes. Let cool for 20 to 30 minutes. Drizzle with caramel sauce, sprinkle with flaky sea salt, and serve warm.