Position a rack in the center of the oven and heat to 325°F (165°C). Line two sheet pans with parchment paper.
Place the finely chopped peanuts in a shallow bowl. If your peanuts aren’t freshly roasted, toast them in a dry skillet over medium heat for 2–3 minutes until fragrant, then cool completely before using.
In a medium bowl, whisk together the flour, baking soda, and fine sea salt.
In a large bowl with a hand mixer (or stand mixer), beat the softened butter, peanut butter, and brown sugar on medium-high until light and creamy, about 2 minutes. Beat in the vanilla and egg just until smooth, about 30 seconds.
Add the dry ingredients to the bowl and mix on low until a soft dough forms and no dry spots remain. Chill the dough for 15 minutes to firm slightly for easier rolling.
Scoop dough into 1-tablespoon portions (about 20 g), roll into smooth balls, then coat each one in the chopped peanuts. Arrange on the prepared sheets, spacing about 2 inches apart.
Use a 1/2-teaspoon measuring spoon to press a deep dimple into the center of each ball before baking. Bake 12–14 minutes, rotating the pans halfway, until the edges look set and the bottoms are pale golden.
Immediately after baking, re-press the centers to reinforce the wells. Cool on the sheet for 5 minutes, then transfer to a rack to cool completely.
Make the ganache: Warm the cream and espresso powder in a small saucepan over low heat until steaming (do not boil). Pour over the chocolate chips in a heatproof bowl. Let stand 2 minutes, then stir until smooth and glossy. Let thicken at room temperature for 10–15 minutes to a spoonable consistency.
Fill and finish: Spoon or pipe the ganache into the centers of the cooled cookies. Sprinkle each with a tiny pinch of flaky sea salt, if using. Let stand until set, about 30 minutes.