Preheat oven to 350°.
Combine pecans and butter, and toss well. Spread mixture evenly onto a jelly-roll pan. Bake at 350° for 12 minutes, stirring once. Remove from oven; sprinkle with salt.
Combine the sugar, water, and corn syrup in a medium saucepan; bring to a boil over medium heat. Cover and cook 3 minutes. Uncover and continue cooking, without stirring, until a candy thermometer registers 250° (about 8 minutes). Remove from heat.
Beat egg whites in a large mixing bowl with a heavy-duty stand mixer at high speed until stiff peaks form; beat in vanilla. Pour hot syrup in a thin stream over beaten egg whites while continuing to beat at high speed. Continue beating until mixture becomes stiff (about 3 minutes). Working quickly, drop by rounded teaspoonfuls onto wax paper. Gently press a pecan half on top of each piece of candy. Store divinity in an airtight container.