Preheat oven to 350 degrees. Whisk together flour, baking powder, baking soda, and salt. Beat peanut butter and butter with a mixer on medium speed until smooth. Add sugars, and beat until pale and fluffy. Add egg and vanilla, and beat until incorporated. Reduce speed to low. Add dry ingredients, and mix until combined.
Scoop level tablespoons of dough, and form into balls. Roll each ball in granulated sugar, and transfer to parchment-lined baking sheets, spacing about 2 inches apart.
Bake until cookies are puffy, about 10 minutes. Remove from oven, and either add candies or make indentations in centers by pressing with the handle end of a wooden spoon. Return to oven, and bake until edges are golden, 6 to 7 minutes more. Transfer sheets to wire racks, and let cool completely.
If using jam, heat in a small saucepan, stirring about 30 seconds. Spoon about 1/2 teaspoon into each indentation. Cookies can be stored for up to 1 week.