Butter the inside of each ramekin thoroughly, then dust them with flour. Tap out any excess flour to help cakes release cleanly after baking.
In a heatproof bowl over simmering water or in short bursts in the microwave, melt 1 cup semi-sweet chocolate chips with 1/2 cup unsalted butter. Stir until smooth and glossy. Set aside to cool slightly.
In a separate bowl, whisk together 2 large eggs, 2 large egg yolks, and 1 cup powdered sugar until pale and slightly thickened.
Slowly add the melted chocolate and butter mixture into the eggs and sugar, whisking continuously to prevent cooking the eggs. Stir in 1 teaspoon vanilla extract.
Sift 1/2 cup all-purpose flour over the batter, then gently fold it in using a spatula. Do not overmix.
Divide the batter evenly among the prepared ramekins. Place on a baking sheet and bake in a preheated 425°F (220°C) oven for 12-14 minutes. Edges should be set but center soft and molten.
Let cakes cool about 1 minute, run a knife around edges to loosen, invert each ramekin onto a plate, and gently lift to release cake. Serve immediately.