In a large bowl, combine milk, water and yeast. Stir to blend. Let stand until yeast is foamy, 5 to 10 minutes. Stir in granulated sugar, vanilla, and salt. Whisk in butter pieces and eggs until eggs are well blended (any remaining lumps of butter will disappear with further mixing).
Beat in 2 ½ cups of the flour. Beat in enough of the remaining flour, about 2 ½ cups, to make a manageable dough that holds together in a rough mass. Turn out onto a floured work surface and knead 2 minutes. Let dough rest 10 minutes. Resume kneading, adding more flour if dough is sticky, until dough is smooth and elastic. Place in an oiled bowl, cover with a towel, and let rise in a draft-free place until dough is doubled in bulk, about 1 hour.
Meanwhile, make the glaze: Preheat oven to 350°. Spread pecans on a baking sheet or in a shallow pan. Bake, stirring once or twice, until lightly browned and fragrant, 7 to 10 minutes. Let cool slightly, then chop coarsely.
In a large saucepan, combing brown sugar, butter, corn syrup, and cinnamon. Cook over medium heat, stirring, until mixture is smooth and bubbles begin to appear around edges of pan. Stir in vanilla.
Butter two 8-inch-square baking pans. Spread 1 cup glaze in each pan; top each with 1 cup toasted pecans. Stir the remaining cup pecans into glaze in saucepan. Set aside.
Punch down dough. Turn onto a lightly floured work surface; divide into 2 equal portions. Roll or pat 1 portion into a 9- by 14-inch rectangle. Spread dough with half the remaining glaze. Roll up jelly-roll fashion, beginning with the long edge; pinch seam to seal. With a serrated knife, gently cut roll into 9 slices, each about 1 ½ inches thick. Place slices, cut side down, in one of the prepared pans. Repeat with remaining dough and glaze.
Let rise, uncovered, until dough is not quite doubled, 40 to 60 minutes.
Preheat oven to 375°. Place a foil-covered baking sheet on a rack just beneath the sticky buns to catch any glaze that might drip. Bake until buns are puffed and golden and glaze is hot and bubbly, 40 to 50 minutes. Let cool in pan up to 5 minutes. Invert each pan onto a large, heatproof serving platter; wait a few seconds, then carefully lift off hot pans. Serve warm.