Preheat oven to 400°F. In a large bowl, combine the flour, baking powder, baking soda, and salt, then set aside.
In a separate bowl, whisk together the melted butter and brown sugar.
Whisk in the eggs and vanilla extract.
Stir in the milk until smooth.
Whisk in the peanut butter until creamy and well-combined.
Fold in the dry ingredients, mixing until just combined. Add the peanut butter chips and fold in, being careful not to overmix.
Line a 12-count muffin tin with muffin liners or grease a muffin pan heavily. Fill each muffin cup most of the way, and sprinkle with coarse sugar (if using).
Bake in the preheated oven for 5 minutes, then turn the temperature down to 350°F. Bake for 14-15 more minutes until the centers are completely set, verified with a toothpick inserted in the center.
Allow the muffins to cool, then serve!