Preheat the oven to 350°F (175°C). Line 2 baking sheets with parchment paper.

In a medium bowl, whisk together the flour, cocoa powder, baking powder, and salt until well combined. Set aside.

In a large mixing bowl, beat the softened butter and granulated sugar with a hand mixer or stand mixer on medium speed until light and fluffy, 2–3 minutes.

Add the egg, egg yolk, and vanilla to the butter mixture. Beat on medium speed until smooth and well combined, scraping down the bowl as needed.

Add the dry ingredients to the wet ingredients in two additions, mixing on low speed just until combined. If the dough seems crumbly or too stiff to roll, mix in 1–3 tablespoons of milk, 1 tablespoon at a time, until the dough holds together smoothly and feels soft but not sticky.

Cover the bowl and chill the dough in the refrigerator for 20–30 minutes, just until slightly firm and easy to handle.

Scoop about 1 tablespoon of dough (about 20 g) and roll it into a smooth ball between your palms. Place the balls on the prepared baking sheets, spacing them about 2 inches apart.

Using your thumb or the back of a small measuring spoon (1/2 teaspoon), gently press a deep indentation into the center of each dough ball. If the edges crack, lightly pinch or smooth them back together.

Bake one sheet at a time for 10–12 minutes, until the cookies are set around the edges but still soft in the centers. The indentations may puff slightly.

Immediately after removing the cookies from the oven, use the back of the measuring spoon again to gently press and redefine the center wells while the cookies are still hot. Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.

While the cookies cool, make the chocolate filling. Place the chopped chocolate in a heatproof bowl. In a small saucepan, heat the heavy cream over medium heat just until it begins to steam and small bubbles form around the edges; do not boil.

Pour the hot cream over the chocolate. Let sit undisturbed for 2–3 minutes to soften, then add the butter, vanilla, and a pinch of salt. Stir slowly with a spatula or whisk until the mixture is completely smooth and glossy.

Let the chocolate filling cool at room temperature for 10–15 minutes, stirring occasionally, until it thickens slightly but is still pourable and spoonable.

Spoon about 1 teaspoon of the chocolate filling into the center of each cooled cookie, letting it pool and level in the thumbprint. If desired, lightly tap the baking sheet on the counter to help the filling settle into a smooth surface.

While the chocolate is still soft, sprinkle the tops with flaky sea salt, chopped nuts, or colored sugar if using.

Allow the filled cookies to sit at room temperature for 1–2 hours, or until the chocolate centers are set but still soft to the touch. Store the cookies in an airtight container at room temperature for up to 4 days.