Preheat the oven to 350°F. Line a 9-inch square baking pan with parchment paper, leaving an overhang on two opposite sides to lift the brownies out later.
Melt 1/2 cup unsalted butter (one 4 oz. stick) (microwave or stovetop). In a medium bowl whisk the melted butter with 1 cup Nutella until smooth. Set this mixture aside and reserve the remaining 1/4 cup Nutella for later.
In a large bowl whisk 4 large eggs and 1 cup granulated sugar (200 grams) for about 2 minutes, until the mixture is slightly thickened and pale yellow.
With the eggs and sugar whisking, slowly add the Nutella-butter mixture a little at a time, whisking constantly until it is fully incorporated.
Add 1/2 cup gluten-free 1:1 flour (60 grams), 1/2 cup unsweetened cocoa powder (43 grams), 1/2 teaspoon salt, and 2 teaspoons pure vanilla extract to the batter. Whisk until just combined and no dry streaks remain (do not overmix).
Pour the batter into the prepared pan and spread it evenly.
Dollop the reserved 1/4 cup Nutella over the top of the batter in several spoonfuls. Use a chopstick, skewer, or the tip of a butter knife to swirl the Nutella through the top of the batter.
Bake at 350°F for 28–33 minutes, until the edges are set and the center is just set (a toothpick inserted near the center should come out with a few moist crumbs).
Remove the pan from the oven and let the brownies cool completely in the pan.
Use the parchment overhang to lift the cooled brownies from the pan. Cut into squares on a cutting board with a serrated knife, wiping the knife clean between cuts for neat slices.