Preheat oven to 350ºF (180ºC).
Butter or spray an 8-inch square baking pan. You can line it with a strip of parchment paper, covering the pan's bottom and two sides. The other two sides will remain unlined but greased. This will help you remove the brownies from the pan.
Melt the chocolate with the butter in the microwave (at 10-15 second intervals, stirring well each time) or in a double boiler with the bottom of the bowl not touching the water.
Add brown sugar and mix well.
Add eggs and combine until it’s creamy. Don't beat; use a whisk.
Add the flour and salt and mix.
Add the peppermint and vanilla extracts and integrate well until smooth and shiny.
Pour batter into the prepared pan. Smooth the top.
Bake for 20-25 minutes, until a bit puffed and the surface is dry. They still have to be a little jiggly in the middle, and a cake tester or toothpick will come out with very moist (but not wet) crumbs attached.
Let cool completely on a wire rack. Run a smooth-bladed knife around the unlined edges to unstuck any parts before lifting the paper carefully so the brownie block comes out in one piece. Place on a flat surface and cut them into squares.
I cover them in plastic wrap or an airtight container and leave them at room temperature for a day before cutting them. It mellows the flavors.