Step 1. Heat oven to 325°F. Grease an 8-inch-by-8-inch baking dish, line with parchment paper (allow a few inches to hang over the edges), and grease paper. Meanwhile, in a small saucepan over medium-high heat, melt butter. Cook, stirring until butter bubbles and starts to brown, 3 to 5 minutes.
Step 2. Remove butter from heat and pour into a medium mixing bowl. Add chocolate and let stand 1 minute, then whisk together. Whisk in sugar, vanilla, eggs, and salt. Fold in cocoa powder and flour until just combined.
Step 3. Pour batter into prepared dish and bake 30 to 40 minutes, or until a toothpick inserted in the center comes out clean. Cool to room temperature.
Step 1. While brownies are cooling, add chèvre and butter to the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed until completely blended, about 2 minutes. Turn mixer down to medium-low, then slowly add sugar and matcha powder. Return mixer to medium speed and beat until completely blended, about 2 minutes. Beat in vanilla on low speed until combined.
Step 1. Top brownies with ½ cup to 1 cup frosting to taste. Sprinkle with dark chocolate and refrigerate 15 to 20 minutes until frosting is firm. Slice and serve.