Preheat an oven to 375º F.
Defrost the puff pastry and roll out. Form a 12-inch square on a baking sheet. Poke it evenly all over with a fork and bake until a light golden, about 15-20 minutes.
While the pastry cooks, peel and finely chop the shallot. Melt the butter in a small pan over medium heat. Add the shallot and cook, stirring frequently, until softened and starting to brown, about 10 minutes.
Meanwhile, cut off and discard the bottom ends of the asparagus so the spear will fit in the pastry, leaving a border of less than an inch. Use a vegetable peeler to peel a bit off the bottom of each spear (this helps them to sit straight on the pastry). Set aside.
Grate the cheese and set aside.
Spread the cooked shallot over the crust. Top with an even layer of the grated cheese. Arrange the asparagus spears on top. Bake until the asparagus is tender and the pastry is browned, about 25-30 minutes.
Let the tart cool, cut into pieces, and serve warm or at room temperature. Like most things made from puff pastry, this tart is best if served warm.