1Heat oven to 350°F. Place pecan pieces on a baking sheet and bake until nuts are lightly toasted and brown, about 10 to 12 minutes. Remove from oven and set aside to cool.
2Butter and lightly flour a 13-by-9-inch baking dish; set aside. Combine toasted pecans and flour in a medium mixing bowl until pecans are well coated; set aside.
3Create a double boiler by filling a pot with 2 inches of water and bringing to a simmer over medium-low heat. Combine butter and chocolate in a heatproof bowl and set over the pot. (Do not let the water touch the bottom of the bowl.) Stir continuously until mixture is completely melted and smooth. Remove from heat and set aside to cool.
4Combine eggs, sugar, and vanilla in the bowl of a stand mixer fitted with a whisk attachment and beat until thick and pale. Add cooled chocolate mixture and beat until evenly combined. Stir in pecan mixture until just incorporated and no white streaks remain.
5Pour into the prepared dish and bake until a toothpick inserted into brownies comes out clean, about 25 to 30 minutes. Let cool before cutting.