Make the ganache first: Combine the chopped dark chocolate and heavy cream in a small saucepan over low heat, stirring frequently until about three-quarters of the chocolate is melted. Remove from heat and whisk until completely smooth, then whisk in the cherry preserves until glossy. Scrape into a bowl, cover, and refrigerate while you prepare the cookies.
Brown the butter: In a medium saucepan, cook the butter over medium heat, stirring, until the milk solids turn deep golden and it smells nutty, 5–7 minutes. Immediately pour into a large mixing bowl, scraping in the browned bits, and let cool until opaque and just warm, 10–15 minutes.
Whisk the dry ingredients in a separate bowl: flour, cocoa, espresso powder, baking powder, and salt.
Build the wet mixture by hand: Whisk the brown sugar into the cooled brown butter until thick and shiny, 30–45 seconds. Whisk in the vanilla and almond extract. Add the egg and then the yolk, whisking just until smooth after each addition.
Switch to a spatula and fold the dry ingredients into the wet until no dry streaks remain. The dough will be soft but not sticky. Let it rest at room temperature for 10 minutes to hydrate the flour.
Portion and chill: Scoop level 1-tablespoon portions (about 30 pieces), roll into smooth balls, and set on a parchment-lined sheet. Refrigerate until firm around the edges, 25–30 minutes.
Preheat the oven to 355°F (180°C) and line two large baking sheets with parchment. Arrange 12–15 dough balls per sheet with at least 2 inches between them.
Create the wells and start baking: Press the center of each ball with a 1/2-teaspoon measuring spoon or your thumb to form a shallow indentation. Bake for 6 minutes, then quickly remove the sheet and press the centers again to deepen the wells.
Return to the oven and bake until the edges look set and the tops are matte but still soft, 2–4 minutes more (total 8–10 minutes). Repeat with remaining trays.
Cool the cookies on the sheet for 5 minutes, then transfer to a wire rack to cool completely before filling.
Fill: Stir the chilled ganache. If it’s too firm, warm it in 5–10 second microwave bursts until spoonable. Spoon or pipe about 1 to 1 1/2 teaspoons into each cookie well.
Set and serve: Let the filled cookies stand at cool room temperature until the centers set, about 30 minutes, or chill for 10–15 minutes for a quicker set.