In a large bowl beat together the butter, sugars, and peanut butter until fluffy.
Mix in the egg and vanilla.
With the mixer on low speed, carefully beat in the flour, baking soda and salt. Then mix in the oats.
If baking the cookies immediately: Preheat the oven to 350F degrees. Form the dough into balls about 1 - 1.5 tablespoons in size.
Place 2 inches apart on a lined cookie sheet. Do not flatten. Then bake for 8-10 minutes or until the tops are just set.
If chilling the dough for thicker cookies: Cover the bowl with plastic wrap and place in the fridge for 2 hours or overnight.
When ready to bake, preheat the oven to 350F degrees. Remove the dough from the fridge and let warm up for 5-10 minutes.
Form the dough into balls about 1 - 1.5 tablespoons in size and place 2 inches apart on a lined cookie sheet.
Bake for 8-10 minutes or until the tops are just set.