Preheat the oven to 425℉, then grease 4 (6-ounce) ramekins with softened unsalted butter. Coat the inside of each ramekin with cocoa powder, tapping out the excess, and evenly space them on top of a baking sheet.
Add the butter and chopped chocolate in a small saucepot over medium heat (you can also use a microwave, melting everything in 30-second intervals). Stir with a rubber spatula until melted and smooth, then remove the pot from the heat and set aside to cool.
Place the eggs, egg yolks, powdered sugar, and vanilla in a medium bowl. Whip everything together with a hand mixer or stand mixer fitted with the whisk attachment on medium speed for 2-3 minutes, until pale, thick, and ribbony.
Once no longer hot, scrape the chocolate ganache into the whipped egg mixture and fold until evenly combined with no streaks. Gently stir in the flour, salt, and espresso powder (if using), folding until just combined.
Evenly divide the batter between each ramekin. At this point, the ramekins can be covered and refrigerated for up to 1 day before baking.
Bake the lava cakes for 10-15 minutes, until the edges are firm and the top is matte, set, but still runny. A toothpick inserted in the edge of the cake should come away clean, but one inserted into the middle should be coated in chocolate. Lava cakes baked from the fridge may need 1-2 more minutes' cook time.
Allow the cakes to cool for 1-2 minutes before running a knife along the edges to loosen them. Carefully invert each ramekin onto serving plates. Lightly dust with powdered sugar and serve immediately with vanilla ice cream or whipped cream.