Preheat oven to 350°F. Line an 8x8-inch pan with parchment paper, leaving a 1-inch overhang on each side.
Place butter in a medium bowl and microwave in 30-second intervals until fully melted. Add the chocolate to the bowl and let stand 3 minutes. Whisk until smooth.
Meanwhile, add eggs and brown sugar to a separate large bowl. Whisk vigorously for at least 3 minutes or until the sugar granules have dissolved completely. You can test this by dipping your finger into the mixture and rubbing it between 2 fingers. You shouldn't feel any sugar granules. The longer you whisk, the more sugar will dissolve.
Whisk in the powdered sugar, cocoa powder, vanilla, and salt. Beat in the chocolate and butter mixture until just combined.
Switch to a rubber spatula or wooden spoon. Fold in the flour until no streaks remain. Fold in the chocolate chips.
Transfer batter to prepared baking pan, and bake 30 minutes. Remove from oven and immediately bang the pan against the countertop a few times (I place a pot holder on the counter when I do this).
Allow the brownies to cool to room temperature, about 90 minutes to 2 hours. The brownies will be ready to slice when you can easily lift them from the pan using the parchment paper overhang and they remain quite firm in the middle (don't bend too much).
Slice the brownies into 16 bars and sprinkle with sea salt (if desired).