Adjust oven rack to middle position and preheat oven to 325°F (160℃). In a medium bowl, whisk flour, baking powder, and salt until well combined; set aside.

In the bowl of a stand mixer fitted with the paddle attachment, cream butter and sugar on medium speed until thoroughly mixed, lightened in color, and fluffy, scraping down sides of bowl and paddle as needed, about 5 minutes. Add egg white and vanilla and mix until fully incorporated. Reduce mixer speed to low, and with mixer running, add half of the flour mixture, and mix until incorporated, then return to medium speed. With mixer still running, slowly pour in evaporated milk. Again, reduce mixer speed to low and add the remaining flour mixture, then return to medium speed and mix until just combined and a thick batter is formed.

Line 2 rimmed baking sheets with parchment paper, then scoop out 1 rounded tablespoon portions of batter, swirling to flatten into 2-inch discs, spaced about 1 ½ inches apart (12 cookies per sheet tray). Using wet fingers, gently press to smooth.

Bake one sheet tray at a time until light brown on edges, 10 to 15 minutes. Let sit on baking sheets until completely cool, about 30 minutes. Once cooled, flip cookies flat side up on baking sheets, ensuring space between them for fudge to overflow.

In a medium saucepan (preferably nonstick), combine sugar, evaporated milk, butter, and corn syrup and cook over low heat, stirring frequently, until butter and sugar are melted and ingredients combine, about 8 minutes. Increase heat to medium and cook until mixture starts to bubble, monitoring temperature with a candy thermometer. When temperature reaches just between 235℉ (113℃) to 240℉ (115℃), turn off heat.

Allow to cool to 210°F (100℃), then add chocolate, cocoa, vanilla, and salt. Stir until thick and glossy, adding very small amounts, ½ teaspoon water at a time, if needed, to keep the fudge glossy and pourable.

Working with 1 cookie at a time, spoon 1 tablespoon fudge onto each cookie, and spread over the entire surface of the cookie. Let fudge topping cool to room temperature and consistency of thick ganache, at least 30 minutes or let firm up to fudge-like consistency overnight. Serve.