Preheat the oven to 425 degrees F. Put the beets in a small baking dish and add about 1/2 inch water. Cover with foil and bake until the beets are easily pierced with the tip of a knife, 1 to 1 1/2 hours. Transfer the beets to a plate and let cool. Wipe off the skins with a paper towel and slice into small wedges; set aside.
Meanwhile, heat the butter in a medium skillet over medium heat. Add the onion, 2 tablespoons water, 1 teaspoon thyme, a big pinch of salt and a few grinds of pepper and cook, stirring occasionally, until the onion is tender and caramelized, 20 to 25 minutes. (Add 1 to 2 teaspoons more water to the skillet, if needed.) Transfer the onion mixture to a food processor and puree until smooth. Let cool.
Roll out the puff pastry into a 9-by-11-inch rectangle on a sheet of parchment paper. Transfer the dough on the parchment to a baking sheet. Score a 1-inch border around the edge of the dough with a paring knife. Brush the border with the beaten egg. Spread the onion puree inside the border.
Combine the beets, asparagus, olive oil, the remaining 1 teaspoon thyme, 1/2 teaspoon salt and a few grinds of pepper in a bowl and toss to coat. Arrange on top of the onion puree.
Bake the tart until the dough is well browned and the asparagus is tender, 30 to 35 minutes. Scatter the goat cheese and chives on top. Serve warm or at room temperature.