In the bowl of an electric mixer, using the paddle attachment, cream the butter, peanut butter, and both sugars until light and well blended.
Combine eggs and vanilla in a liquid measuring cup. Add eggs one at a time to electric mixer, blending well before adding the next.
In a medium bowl, whisk together flour, baking powder, and salt.
On low speed, add to egg and butter mixture until a dough forms.
Scoop dough into a smaller bowl. Cover with plastic wrap and chill for approximately 1 hour.
Preheat oven to 375 degrees Fahrenheit.
Using a 2-ounce or ¼ cup ice cream scoop, scoop dough and roll into balls. Place balls on a parchment-lined sheet pan, leaving 2 inches between cookies.
Place vegetable oil on a small plate. Dip a large fork into the oil and press in a crisscross pattern onto each ball of dough. Sprinkle generously with sea salt.
Bake for approximately 14-16 minutes, or until cookies are light brown. They will continue to set up as they cool.