Preheat the oven to 350°F. Line large baking sheets with parchment paper or silicone baking mats.

In a medium bowl, whisk together the flour, salt, baking powder, and baking soda.

In a large bowl, use an electric mixer to beat the butter, sugar, and cocoa powder on medium speed until well combined, about 2 minutes. Add the eggs, one at a time, and beat until well combined. Add the vanilla and continue mixing on medium speed until mixture is smooth, about 1 minute. Add the melted and cooled chocolate and beat until just blended. On low speed, gradually add the flour mixture and beat until combined. Dough may be sticky and wet.

Using a 1 1/2-tablespoon scoop, drop rounds of dough onto prepared baking sheets, spacing at least 1 1/2-inched apart. Bake until the cookies are puffed and the tops are cracked, 11 to 13 minutes. As soon as the cookies are out of the oven, use the rounded side of a teaspoon or the end of a thick-handled wooden spoon to make a deep well in the center of each cookie. Let the cookies sit on the cookie sheet for 5 minutes before transferring to a wire rack to cool completely.

Melt the chocolate and butter in the microwave in 30 second bursts, stirring between each burst, until the mixture is melted and smooth. Be careful not to overheat. Set aside until cool and slightly thickened.

Spoon the chocolate mixture into the wells of the cooled cookies. Set aside until the chocolate filling firms up, about 1 hour. Serve immediately or store in an airtight container for up to 2 days.