In the bowl of a stand mixer, beat the butter on medium speed until light and fluffy, about a minute. With the mixer on low, add the vegetable oil, sugar, and vanilla slowly and in succession after. Add the eggs one at a time and beat until well incorporated.
In a separate bowl, whisk together the flour, baking soda, and salt. Add to the wet ingredients in increments while the mixer mixes on low speed, until combined.
Add the 2 tablespoons of sprinkles and mix until just combined. Refrigerate the dough for at least an hour, or pop in the freezer for 15-20 minutes.
Roll the chilled dough into heaping tablespoon-sized balls and flatten just slightly with the palm of your hand. Bake in an oven heated at 350 degrees Fahrenheit for about 12 minutes, or until the center is just set and the edges are lightly golden. Let cool completely.
To make the buttercream, beat the butter until light and fluffy on medium speed in the bowl of a stand mixer fitted with the paddle attachment. Turn it to low and add the vanilla.
With the mixer running, add the powdered sugar in ½ cup portions at a time. Keep an eye on the buttercream - you may not need all 2 1/2 cups, or you may want a bit more to reach your desired consistency.
With the mixer running on low, add the tablespoon of sprinkles and mix until interspersed. Add the buttercream to a piping bag with a wide circle tip (or you can use a plastic bag and simply snip a corner). Evenly pipe the buttercream just inside the edge of the underside of a cookie. Gently press it together with the underside of an un-iced cookie.