Sift together the flour, baking powder, fine sea salt, and baking soda into a bowl. Set aside.

In the bowl of a stand mixer fitted with the paddle attachment (or in a large mixing bowl, using a handheld mixer), cream the butter, both sugars, and the cane syrup together on medium speed until superlight and fluffy, 3 to 5 minutes.

Add the peanut butter, egg, and vanilla and mix until completely incorporated. Turn the speed down to low and gradually add half of the flour mixture, remaining flour, mixing until just combined. Gradually add the remaining flour, mixing until just combined, then add the peanuts, mixing until incorporated.

Remove the bowl from the mixer stand (if using) and finish mixing by hand to make sure no bits of flour or butter are hiding on the bottom of the bowl and the dough is thoroughly mixed.

Cover the bowl with plastic wrap and chill the dough in the refrigerator until slightly firm, about 30 minutes.

Position the racks in the middle and lower third of the oven and preheat the oven to 325°F (165°C).

Line two baking sheets with parchment.

Use a small ice cream scoop or a tablespoon to form the cookies (about 1 rounded tablespoon each) and place on the prepared baking sheets, leaving 2 inches (5 cm) between them to allow for spreading.

Lightly touch each cookie with the tines of a fork to make a crisscross pattern (keeping it old-school). Top each cookie with a pinch of flaky sea salt. (Able to freeze cookies on cookie sheet until firm, then transfer to ziplock bags and freeze up to 2 months. Sprinkle with salt just before baking.)

Bake the cookies for 10 to 12 minutes, rotating the pans halfway through and switching their positions, until golden brown.

Let cool completely on the pans or wire racks.

The cookies can be stored in an airtight container at room temperature for up to 3 days.