Lightly grease two 6-ounce ramekins and place them on a baking sheet.
Preheat oven to 350°F (177°C).
Coarsely chop 2 ounces of the chocolate. Combine with heavy cream in a heatproof bowl and microwave in 20-second intervals, stirring until smooth.
Whisk in brown sugar until dissolved, then whisk in oil, beaten egg, and vanilla.
In a small bowl, whisk flour, baking powder, and baking soda. Fold into the wet mixture until just combined.
Divide batter evenly between ramekins, filling about 3/4 full.
Cut the remaining 1 ounce of chocolate in half. Place one piece into the center of each ramekin and gently submerge.
Bake for 15–17 minutes, rotating the pan halfway through.
Serve warm for the best gooey chocolate center.