Place the flours into the bowl of a food processor fitted with the blade. Add the cold diced butter and pulse until the mixture resembles coarse sand with some larger pea-sized pieces. Add two tablespoons of the ice water and process for a moment. Test to see if the dough holds together when you squeeze a bit with your hand. If not, add the remaining water and process again. Use your hands to quickly form the dough into a ball. Then flatten into a disk between two approximately 14-inch square pieces of parchment paper. Wrap in the parchment followed by a gallon-sized zip-top bag, and refrigerate for at least an hour (up to a week) before proceeding.

Slice off and discard roots and dark green parts from leeks. Slice white and light green parts in half lengthwise and then cut each length crosswise into thin half-moons. Wash and dry very well in a salad spinner to remove all grit. Melt butter over medium heat in a medium nonstick skillet. Add leeks and salt and cook, stirring frequently, until jammy and tender, about 10 minutes.

Preheat oven to 400° F with a rack in the center. While the leeks finish cooking, sprinkle a little bit of flour on each side of the dough and then replace the parchment. Using a rolling pin and turning the dough a quarter-turn clockwise every so often, roll the dough out between the parchment into a 14-inch circle. Remove the top sheet of parchment and slide the dough onto a pizza pan or baking sheet. Brush the dough with the olive oil, leaving a 1 ½-inch border around the outside. Spread the ricotta evenly over the dough, still leaving a plain 1 1/2-inch border as you add all ingredients. Sprinkle half the grated cheese evenly over ricotta. Spoon the warm leeks over the cheeses in an even layer. Lay asparagus evenly over the leeks. Sprinkle with remaining grated cheese. Working in 6 to 8 sections, fold the outer ring of dough over the filling to form a crust, pressing gently to adhere the sections together. In a small bowl, beat together the egg with the milk or cream. Brush egg wash over outer ring of crust.

Bake for 40 minutes, until nicely browned. While tart is hot, cut or tear burrata into pieces and distribute over filling. Drizzle with balsamic glaze and sprinkle with flaky salt and freshly ground black pepper. Cut into wedges and serve warm or at room temperature.