Heat the oven to 350°F / 175°C with a rack in the center. Lightly butter an 8x8-inch metal pan and line it with parchment paper, leaving some overhanging on two sides to lift the brownies out later.
Put the butter and chopped chocolate in a large heatproof bowl. Set it over a pan of barely simmering water, without the bowl touching the water, and stir until melted and smooth, about 5 minutes. Alternatively, microwave in 30-second bursts, stirring between each, until just melted.
Take the bowl off the heat and whisk in the sugar. The mixture will look slightly grainy. Let it cool for about 5 minutes, until warm rather than hot.
Add the eggs one at a time, whisking well after each. Add the vanilla, then whisk vigorously for about 1 minute until the batter is thick, smooth and glossy.
Sift the flour, cocoa powder and salt over the bowl. Fold gently with a spatula until no dry streaks remain, then fold in the chocolate chips.
Scrape the batter into the prepared pan and spread it into an even layer.
Bake for 25 to 30 minutes, until the top is shiny and set, the edges are firm and a toothpick inserted 1 inch from the center comes out with moist crumbs (not wet batter). Start checking at 25 minutes.
Cool the brownies completely in the pan on a wire rack, at least 1 hour. Lift them out using the parchment and cut into 16 squares with a sharp knife, wiping it clean between cuts.