Heat the oven to 325°F (163°C). Grease a 9-inch springform pan and line the bottom with parchment; lightly grease the parchment as well.
Bloom the chocolate: In a small saucepan over medium heat, melt the butter until hot and just starting to steam. Remove from heat, immediately add the chopped bittersweet chocolate, cocoa powder, and espresso powder, and let sit 2 minutes. Whisk until completely smooth and glossy.
In a large bowl, whisk the granulated sugar, brown sugar, eggs, and vanilla for 60–90 seconds until thickened and slightly lightened in color.
Stream the warm chocolate-butter mixture into the egg mixture while whisking until fully combined.
Sift the flour, fine sea salt, and baking powder over the bowl. Switch to a spatula and fold just until you no longer see dry spots. Fold in the milk chocolate chips. Let the batter rest 10 minutes to hydrate for extra fudginess.
Scrape the batter into the prepared pan and smooth the top. Tap the pan firmly on the counter twice to release large air bubbles.
Bake 32–38 minutes, until the edges are set, the top is shiny and cracked, and a toothpick inserted 2 inches from the edge comes out with damp crumbs (the very center may still appear slightly soft).
Cool the cake on a rack for 20 minutes, then run a thin knife around the edge and unlatch the springform ring, keeping the cake on the base.
Make the ganache: Warm the heavy cream in a small saucepan or microwave until steaming but not boiling. Pour over the chopped milk chocolate and butter in a bowl, cover, and let stand 2 minutes. Add a pinch of salt, then stir from the center outward until silky smooth. If any bits remain, microwave in 10-second bursts, stirring between each, until melted.
Spread the ganache over the warm cake, nudging it to the edges so it gently drips. Let stand 15 minutes to set, then sprinkle with flaky sea salt if using. For extra-neat slices, chill 20 minutes before cutting with a hot, dry knife.