Preheat the oven to 350 (F) and grease an 8-inch square baking dish with butter.
In a double boiler (or, a heatproof bowl over a pan of hot water), melt 8 oz. chocolate, using a rubber spatula to stir the chocolate constantly to prevent it from burning.
Once the chocolate has completely melted, add 6 tablespoons unsalted butter, using the rubber spatula to stir the butter into the chocolate.
Once the chocolate and butter have combined, remove the boiler from heat and pour the mixture into a heatproof mixing bowl. Whisk in 3/4 cup light brown sugar, 2 eggs, and 1 teaspoon vanilla until thoroughly combined.
Add 1/4 cup flour to the mixture and use a rubber spatula to fold the flour into the chocolate mixture. Be careful not to overmix or you'll get tough brownies and I will cry for you! Simply incorporate the flour until there are one or two flour streaks left in the batter, but no more than that.
Bake for 35 minutes in the preheated oven, or until the brownies are set and barely firm in the middle. Cool on a cooling rack for 15 - 20 minutes before removing from pan. Brownies will be very melty and delicious.