Preheat the oven to 350°F (177°C). Position the oven racks in the upper and lower thirds and line two 13 x 18-inch (33 x 46 cm) baking sheets with parchment paper. Keep the sheets ungreased so the cookie dough holds its shape during baking.
In a medium mixing bowl, whisk together 1 1/4 cups (150 g) all-purpose flour, 1/3 cup (28 g) unsweetened Dutch-process cocoa powder, 1 teaspoon (2 g) instant espresso powder, and 1/2 teaspoon (3 g) fine table salt. Mix for about 30 seconds, until the cocoa and espresso are evenly distributed and no visible pockets of powder remain.
Place 12 tablespoons (170 g) softened unsalted butter and 1 cup (113 g) sifted confectioners' sugar in a large mixing bowl. Beat with an electric mixer on medium speed for approximately 2 minutes, until smooth, creamy, and slightly lighter in color. Add 1 teaspoon (5 mL) pure vanilla extract and 1/4 teaspoon (1.25 mL) peppermint extract. Beat for another 20 to 30 seconds until evenly incorporated.
Add the dry ingredient mixture to the creamed butter in two additions, mixing on low speed until the flour disappears into the dough. Stop as soon as the mixture forms a cohesive, soft chocolate dough with no dry streaks. Avoid extended mixing because additional gluten development can make the baked cookies firm rather than tender.
Divide the dough into 24 equal portions and roll them into smooth balls. Arrange 12 portions on each prepared baking sheet, spacing them approximately 2 inches (5 cm) apart. Gently press the rounded back of a 1/2-teaspoon (2.5 mL) measuring spoon into the center of each ball to create a well without pressing through the bottom. Smooth any large cracks around the edges so the cookies maintain their shape.
Bake at 350°F (177°C) for 10 to 12 minutes, switching the baking sheets between racks and rotating them halfway through for more even heat exposure. The cookies are ready when their edges look dry and set, their surfaces have lost their wet sheen, and the centers remain slightly soft. If the indentations have puffed up, gently press them back down with the measuring spoon while the cookies are still warm. Let the cookies cool for approximately 30 minutes, initially on the baking sheets and then on a wire rack, until completely cool.
Place 4 ounces (113 g) finely chopped semisweet chocolate in a heatproof bowl. Pour 1/2 cup (120 mL) cold heavy whipping cream into a small saucepan and heat over medium-low heat for approximately 2 to 3 minutes, until small bubbles form around the edges and the cream is steaming but not boiling. Pour the hot cream over the chocolate and let stand for 2 minutes. Stir slowly from the center outward until the mixture becomes smooth, glossy, and evenly combined, with no visible chocolate pieces.
Spoon the ganache into the completely cooled cookie indentations, distributing it evenly without overflowing the wells. Sprinkle the filling with 1/4 cup (approximately 30 g) finely crushed peppermint candy canes, divided evenly. Place the cookies on a refrigerator-safe tray and refrigerate for approximately 30 minutes, until the ganache is set enough to hold its shape when the tray is gently moved.