Preheat the oven to 375°F. Cover the baking sheets with parchment paper to encourage even baking and prevent the cookies from sticking.
Melt the butter. Place the salted butter in a microwave-safe bowl and heat it in 20-second intervals, stirring after each interval, until completely melted. Allow it to cool for 2–3 minutes before combining it with the egg.
In a medium bowl, combine the slightly cooled melted butter, packed brown sugar, granulated sugar, room-temperature egg, and vanilla extract. Mix vigorously for approximately 1 minute, or until the mixture is smooth, thoroughly blended, and somewhat lighter in appearance.
In a separate small bowl, thoroughly whisk the gluten-free flour blend with the baking powder, cinnamon, and baking soda. Continue until the mixture is uniform and free of visible lumps.
Add the dry mixture to the wet ingredients. Fold gently only until the flour mixture is nearly incorporated, taking care not to overwork the dough.
Fold in the gluten-free rolled oats and butterscotch morsels until both are distributed evenly. The finished dough should hold together while retaining visible texture from the oats.
Using a cookie scoop or spoon, place rounded portions of dough onto the prepared baking sheets. Leave approximately 2 inches of space between each portion to accommodate spreading.
Bake medium cookies for 8–9 minutes or large cookies for approximately 13 minutes. Remove them when the edges are pale golden and the centers appear just set but still slightly underbaked.
Leave the cookies on the hot baking sheet for 2–3 minutes. This brief resting period allows residual heat to finish setting their structure.
Carefully transfer the cookies to a wire rack and allow them to cool completely. Their chewy centers, lightly crisp edges, and butterscotch flavor will become more pronounced as they set.