Combine ¾ cup (1.5 sticks) unsalted butter, ¾ cup packed light brown sugar, and ¼ cup granulated sugar together in the bowl of a stand mixer fitted with a paddle attachment. Cream mixture together on medium-high speed for about 4 minutes. Add 1 large egg, and 1 teaspoon vanilla extract and turn mixer on to medium-high speed again. Mix for about 2-3 minutes, until the mixture is very light and fluffy.
Stop mixer and add 2 cups all purpose flour, 3.5 ounce Package instant butterscotch pudding mix, 1 teaspoon baking soda, and ⅛ teaspoon salt. Mix together on low speed until combined.
Add 9-10 oz butterscotch chips and mix on low speed until just incorporated.
Scoop out roughly 2 tablespoon sized amounts of dough and roll into balls about 1"-1.5" in diameter. That will give you roughly 24-30 balls. Flatten balls slightly with the palm of your hand.
Place flattened dough balls on a plate and cover with plastic. Chill in the fridge for at least 2 hours (up to 5 days) so they don't spread when baking. You could also freeze the dough in an airtight container.
Heat oven to 350°F. After chilling, transfer dough patties to a baking sheet lined with parchment. Bake for 11 minutes. Cookies will still look underdone but the edges will be set. Cool on baking sheet for at least 10 minutes before serving.
Notes
Don't try to use "cooked" or sugar free pudding mix. They won't work.