Melt the butter in a heavy medium saucepan over medium heat. Whisk in the flour and cook for 3 minutes, but do not allow the mixture to brown. Add the hot milk and bring to a boil (the sauce will be thicker than most béchamel sauce). Once the sauce comes to a boil continue simmering, whisking constantly, until the sauce thickens, about 5 minutes. Remove the saucepan from the heat and whisk in the cheese. Season the béchamel sauce to taste with salt and pepper. Transfer the blue cheese béchamel to a container and cool, then cover and refrigerate until it is cold, at least 6 hours.
Combine the beet juice, water, sugar, and lemon juice in a heavy medium saucepan. Scrape the seeds from the vanilla bean into the saucepan, then add the bean. Bring the mixture to a boil over high heat, stirring to dissolve the sugar. Add the pears, cover with parchment paper, and simmer over medium heat until tender but not mushy, and rotating the pears occasionally to cook evenly, about 15 minutes. Transfer the pear to a baking sheet to cool. Once the pears are cool, quarter the pears lengthwise.
Combine the vinegar and honey in a heavy small saucepan and bring to a boil over high heat, whisking to blend. Reduce the heat to medium and simmer until the mixture is reduced and syrupy, about 10 minutes. Transfer the reduction to a bowl and set aside to cool.
Preheat the oven to 425°F. Line a heavy large baking sheet with parchment paper. Roll out each piece of the puff pastry dough on a lightly floured work surface into a 11-inch square. Transfer 1 pastry square to the prepared baking sheet. Gently spoon the blue cheese béchamel over the center of the pastry on the baking sheet, leaving a 2-inch border around the edges. Mix the egg yolk with 1 teaspoon of water in a small bowl to blend. Brush the pastry edges with some of the egg wash. Top with the second pastry. Press the edges firmly together to enclose the filling and seal the pastries well, taking care to remove any air bubbles. Using a sharp knife and the bottom of a 10-inch-diameter tart pan bottom as an aid, cut the pastry into an 11-inch-diameter round. Brush the top of the pastry with the egg wash. Using the knife tip, decoratively score the top of the pastry. Bake until the pastry puffs and is golden brown on top, about 35 minutes. Reduce the heat to 350°F and continue baking until the pastry is cooked through and golden brown all over, about 30 minutes.
Meanwhile, stir the remaining 2 tablespoons of water and granulated sugar in a small saucepan over medium-high heat until the sugar dissolves and the mixture is syrupy, about 2 minutes. Brush the warm sugar syrup over the warm tourte (this will make the pastry crispy) and serve.
Cut the warm tourte into 8 pieces and place each piece on a plate. Place a wedge of poached pear alongside the tourte, then drizzle with some of the honey-balsamic reduction and serve.