1Melt the butter in a medium frying pan over medium-low heat and cook until it browns and emits a toasted smell (the butter should be golden brown but not burnt), about 5 minutes. Transfer the butter to a large mixing bowl and let it cool to room temperature.
2Whisk together the flour, brown sugar, baking powder, and salt in a large bowl to aerate and break up any lumps; set aside. Add the eggs, buttermilk, and vanilla extract to the cooled brown butter and whisk until evenly combined. Pour the buttermilk mixture into the flour mixture and stir until the dry ingredients are just moistened through.
3Heat the waffle iron to medium according to the manufacturer’s instructions. Once heated, brush the iron with butter or oil, fill it with batter, close the lid, and cook until the waffle is golden brown, about 4 minutes for Belgian style. Repeat with the remaining batter, brushing the iron with butter or oil between each waffle. Serve immediately with maple syrup and a dollop of plain yogurt.