Preheat the oven to 350 degrees (F) and adjust an oven rack to the middle position. Line a 9 x 13-inch baking dish with parchment, making sure the parchment comes up the sides of the dish.
In a medium bowl, whisk together the gluten-free flour, oats, baking soda, and salt.
Place the 3/4 cup butterscotch chips in a large bowl.
In a small saucepan, melt the butter over medium heat and let the butter cook until golden brown in color, about 10 minutes, taking care not to burn.
Pour the hot butter over the butterscotch chips and whisk together until melted and smooth. Whisk in the brown sugar until combined. Whisk in the egg* and vanilla. Stir in the flour mixture just until incorporated.
Scrape the batter into the prepared pan and smooth the top, spread the batter evenly to the edges.
Mix the flour, oats, sugars, and salt together. Add the melted butter and mix until incorporated. Pat the crumble onto the top of the batter.
Bake the bars until a toothpick inserted into the center comes out with just a few moist crumbs, about 16 - 18 minutes.
While the bars bake, place the 1/4 cup butterscotch chips, brown sugar, cream, and salt in a small microwave-safe bowl. Microwave the mixture until melted and smooth, about 2 minutes on 50 percent power. Whisk to combine well.
Drizzle the glaze over the warm bars. Let the bars cool completely in the pan. Remove the bars from the pan using the parchment overhang and cut into squares. Serve.
*Make sure the butterscotch mixture isn't too hot before adding the egg.