Place oven rack in middle position and preheat oven to 375 degrees F. Line two non-insulated cookie sheets or sheet pans with either parchment paper or silicon baking pads (e.g., Silpad), if desired (*see note). Set aside.

In a mixing bowl, sift together the flour, baking powder, baking soda and salt. Set aside.

In bowl of standing mixer fitted with paddle attachment (or an electric mixer, or by hand), beat butter at medium-high speed until creamy. Add brown sugar and 1 cup of the granulated sugar and beat until light and fluffy, about 3 minutes (or 5 to 6 minutes by hand), stopping to scrape down bowl as necessary. Reduce speed to medium-low and add in peanut butter, beating until fully incorporated, then add eggs, one at a time, beating well after each addition until they are well incorporated; beat in vanilla extract. Reduce speed to lowest setting or gently stir by hand; gradually add in the sifted dry ingredients, scraping sides of mixing bowl as necessary, and beating just until it's incorporated; don't overmix or the cookies will be tough. (The dough should be stiff and slightly sticky -- not dry.)

At this point, for a soft and chewy cookie, the dough should be double wrapped in plastic and refrigerated for 1 to 3 hours. (The dough can keep in the refrigerator for a couple days or stored in the freezer for up to 1 month.) For a thinner and crisper cookie, use dough immediately without chilling.

Shape dough into 1- to 1 1/4-inch sized balls, and roll in remaining 1/3 cup granulated sugar, if desired. (Make them all the same size per baking pan so they bake evenly. A #40 cookie scoop is the perfect size.) Place balls 2-inches apart, in staggered rows (**see note) of 3-2-3-2-3, on prepared baking sheets or regular ungreased pans. Gently press and flatten each dough ball with back of fork dipped in cold water, making a criss-cross design to tops.

For a light, crisp cookie: Place one cookie sheet at a time onto center rack of a preheated 375 degree F oven and bake until just starting to turn golden, about 9 to 10 minutes, rotating baking sheet if needed for even baking. Do not over-bake! (You want the tops to just start taking on color, and still have slightly puffy and moist centers. When removed from the oven, they will flatten out and continue cooking through.) Remove cookie sheet from oven and allow cookies to cool and set for 1 to 2 minutes on baking sheet, then transfer to wire racks to cool completely. If baked on parchment paper or silicon baking pad, slide the entire sheet off the tray directly onto the cooling racks. Meanwhile, you can re-use the cookie sheet immediately for baking the next batch.

For a soft, chewy cookie: Refrigerate the dough for 1 to 3 hours. Once chilled, form balls and flatten with a fork as before. Then, refrigerate uncooked cookies on baking sheet for at least 15 minutes to firm dough back up. Place one cookie sheet at a time onto center rack of a preheated 325 degree F oven and bake until cookies are puffed and just slightly golden along edges, about 15 minutes, rotating baking sheet if needed for even baking. Do not over-bake! Remove cookie sheet from oven and allow cookies to cool and set for 1 to 2 minutes on baking sheet, then transfer to wire racks to cool completely. If baked on parchment paper or silicon baking pad, slide the entire sheet off the tray directly onto the cooling racks. Meanwhile, you can re-use the cookie sheet immediately for baking the next batch.

Store cookies in an airtight container at room temperature for up to 1 week.

Makes about 60 cookies.

*Note: Parchment paper or silicon baking pads are not a necessity but does make for easy cookie removal and cleanup. If using parchment, remove baking sheets from oven and immediately slide cookies on parchment directly onto cooling racks.

**Note: By staggering the rows of cookie batter, it's easy to fit 13 cookies (2-inches apart) on a single large cookie sheet.