1. Spread pecans in a single layer in a 10- by 15-inch pan. Bake in a 350° oven until nuts are fragrant and just slightly darker, about 7 minutes.
2. In a 1- to 1 1/2-quart pan, cook butter over high heat just until it begins to brown and have a nutty aroma, 4 to 5 minutes; remove from heat.
3. In a blender or food processor, finely grind 1/4 cup pecans.
4. In a bowl, beat eggs to blend. Add ground pecans, pecan halves, browned butter, flour, brown sugar, maple syrup, and vanilla. Mix well. Pour nut mixture into the baked crust.
5. Bake in a 350° oven until center is set when pie is slightly shaken, about 25 minutes. Let pie stand at least 30 minutes or until cool.
6. Remove pan rim and cut pie into wedges.