For the easy caramel sauce: Preheat the oven to 350 degrees F. Mix the brown sugar, half-and-half, butter and salt in a saucepan over medium-low heat. Cook, whisking gently, until it thickens, about 5 minutes. Add the vanilla and cook to thicken further, 1 more minute. Remove from the heat and set aside to cool slightly.
For the pumpkin cake: Grease the bottom and sides of a 12-inch skillet with butter or nonstick spray.
In a large bowl, combine the flour, pumpkin pie spice, baking powder, baking soda and salt. Make a well in the center of the flour mixture and add in the butter, granulated sugar, pumpkin, egg and vanilla extract. Mix by hand until fully incorporated. (The mixture will be thick.)
Once combined, press half of the mixture into the bottom of the skillet, making sure the batter goes halfway up the sides.
Pour the cooled caramel over the batter. Drop the remaining pumpkin cake batter over the caramel by the spoonful, covering the caramel as well as you can.
Bake until the edges are firm and golden, 22 to 28 minutes.
For the maple glaze: Meanwhile, bring the maple syrup and butter to a low boil in a small saucepan over medium-high heat, stirring constantly, about 2 minutes.
Remove from the heat and gradually whisk in the confectioners' sugar until smooth. Continue to whisk until the mixture begins to thicken and cool slightly, about 1 minute.
Allow the cake to cool in the skillet for 5 minutes. Pour the maple glaze over the cake and serve immediately.