Cookies:
1½ cups unbleached all-purpose flour, spooned and leveled
½ teaspoon baking soda
½ teaspoon fine sea salt
½ cup unsalted butter, at room temperature
½ cup firmly packed dark brown sugar
½ cup granulated sugar
1 teaspoon pure vanilla extract
¾ cup well-stirred natural, salted crunchy peanut butter
1 large egg, at room temperature
Ganache filling:
3 ounces semisweet chocolate
3 tablespoons unsalted butter
Prepare the cookies:
Position racks to the upper and lower thirds of the oven, and preheat it 350°. Line 2 ungreased cookie sheets with parchment paper.
In a medium bowl, whisk together the flour, baking soda, and salt. In the bowl of an electric mixer fitted with the paddle attachment, beat together the butter, brown and granulated sugar, and vanilla on medium-low speed just until smooth and creamy, about 2 minutes. Add the peanut butter and mix until smooth. Beat in the egg. Reduce the speed to low, and gradually stir in the flour mixture until the dough comes together.
Place level tablespoons of dough about 1½ inches/4 cm apart on the prepared baking sheets, 20 per sheet (they don’t spread much). Using a small measuring spoon with a deep well, or a melon baller, make a 1-inch divot in the center of each cookie. Bake the cookies just until they are firm and turning golden brown at the edges but still very soft in their centers, about 12 minutes, rotating the sheets from front to back and top to bottom about halfway through the baking time. As soon as the cookies come out of the oven, reinforce the indentations with whatever tool you used before. Let the cookies cool for 5 minutes on the pans, then transfer to a wire rack to cool completely.
Prepare the ganache:
Place the chocolate and butter in a microwave-safe bowl. Microwave on HIGH in 30-second intervals, stirring well, until smooth. Let cool slightly. Transfer the ganache to a small resealable plastic bag and work it toward the corner of the bag. Snip off a tiny bit at the corner of the bag with scissors. Fill each divot with ganache. Let the cookies rest at room temperature until the ganache is firm, about 1 hour, or refrigerate to set in about 30 minutes.