Preheat the oven to 350°F (175°C). Grease and flour a 9x13-inch baking pan or line with parchment.
In a medium bowl whisk together the all-purpose flour, baking powder, baking soda, and salt; set aside.
In the bowl of a stand mixer, cream the softened unsalted butter with light brown sugar and granulated sugar on medium speed until light and fluffy, about 3 to 4 minutes.
Add the eggs one at a time, beating well after each addition, then mix in the vanilla extract.
With the mixer on low, alternately add the dry ingredients and buttermilk in three additions, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.
Fold in the butterscotch chips by hand until evenly distributed through the batter.
Spread the batter evenly into the prepared pan and bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs.
While the cake bakes, make the caramel icing: combine 0.5 cup unsalted butter and 1.0 cup light brown sugar in a small saucepan over medium heat. Stir until the butter melts and the sugar dissolves.
Carefully whisk in the warm heavy cream. Bring to a gentle simmer and cook for 2 to 3 minutes, stirring constantly, then remove from heat and stir in vanilla and 0.25 teaspoon salt. Allow the icing to cool slightly so it thickens.
When the cake is done, let it cool in the pan for 10 minutes, then prick the top lightly with a skewer and pour the warm caramel icing over the cake so it absorbs into the crumb. Cool completely before slicing.
Notes
Store leftover cake covered at room temperature for up to 3 days. For a firmer icing, chill briefly before serving.