Preheat the oven to 375°F and line baking sheets with parchment paper.
In a large bowl, cream together the butter, sugar, brown sugar, and dry pudding mix on medium speed for 2 minutes, until light and fluffy.
Beat in the vanilla and eggs until fully incorporated.
Add the baking soda, salt, and flour. Mix just until combined. Do not overmix.
Scoop dough into 3-tablespoon balls. Roll each dough ball in sprinkles until well coated.
Refrigerate dough balls for 30 minutes to control spread and encourage thicker cookies.
Bake for 10–12 minutes, until edges are lightly golden and centers look slightly underbaked.
Cool on the baking sheet for 5 minutes before transferring to a rack.