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Preheat oven to 350 degrees.
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In a large bowl, cream together the butter, peanut butter, sugars, and vanilla.
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Add eggs and beat well.
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Add the flour, baking powder, baking soda, and salt and mix well.
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Add oatmeal and mix until evenly distributed throughout the dough.
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Using a small cookie scoop, place cookies two inches apart on Silpat-lined baking sheets, or ungreased stoneware.
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Bake 9 or 10 minutes, until edges start to brown. Centers will still look soft.
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Remove from oven and let cool slightly on baking sheets before transferring to wire racks to cool completely.
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To make the cream filling, mix the butter, powdered sugar, peanut butter, and milk or cream until smooth. Spread on the bottom half of one cookie, then top with another cookie to make sandwiches.