Line two baking sheets with parchment paper. If your peanuts are not already roasted, toast them in a 350°F (175°C) oven for 5–7 minutes, cool completely, then chop finely.
In a medium bowl, whisk together the flour, baking powder, and fine sea salt until well combined; set aside.
In a large bowl with a hand mixer (or a stand mixer with the paddle), beat the softened butter with the brown sugar and granulated sugar on medium speed until light and creamy, about 2 minutes. Beat in the peanut butter and vanilla until smooth, then mix in the egg just until fully incorporated.
Add the dry ingredients to the bowl in two additions, mixing on low only until the dough comes together and no dry streaks remain. The dough will be soft but not sticky.
Scoop dough into 1 tablespoon portions (about 20 g each), roll into smooth balls, and coat each one in the chopped peanuts. Space on the prepared sheets about 2 inches apart.
Use a 1/2-teaspoon measuring spoon or the end of a wooden spoon to press a shallow well in the center of each ball. Chill the trays for 10–15 minutes to help the cookies hold their shape.
Bake at 325°F (165°C) for 13–15 minutes, until the edges look set and the bottoms are just turning golden. Rotate the pans once if your oven bakes unevenly.
Immediately after baking, gently press the centers again to refresh the wells. Cool on the sheets for 5 minutes, then transfer to a wire rack to cool completely before filling.
Make the ganache: Place the chocolate chips and espresso powder in a heatproof bowl. Heat the cream in a small saucepan over medium until steaming (just below a simmer), then pour it over the chocolate. Let stand 2 minutes, then whisk until glossy and smooth. Allow to thicken for about 10 minutes at room temperature.
Fill the thumbprints by spooning or piping ganache into each well. Finish with a tiny pinch of flaky sea salt. Let the ganache set at room temperature for 30 minutes, or refrigerate the cookies for 10 minutes to speed it up.