Preheat the oven to 180 degrees C. Grease and line with baking paper a 20cm square cake tin.
Melt the butter and 3/4 cup of the chocolate in a bowl set over gently simmering water. Remove from the heat and leave to cool.
Whisk the eggs in a bowl with electric beaters. Gradually add the sugar; beat continuously until the mixture is thick, foamy and leaves a ribbon-like trail when the beaters are lifted. Add the vanilla essence and the chocolate mixture and blend in thoroughly.
Sift flour and cocoa powder over the mixture then scatter in walnuts and the rest of the chocolate. Fold the mixture together with a large spoon.
Pour batter into the prepared tin and bake for about 30 minutes or until the top is a rich brown. Place a piece of foil over the top if it looks to be in danger of burning. Cool brownies briefly in the tin; cut into squares. Cool brownies completely on a wire rack.