Either flown in fresh from Puglia, or made locally by a domestic artisan near you, burrata is becoming more widely available all over the world. Here’s a chance to see the many ways burrata can be utilized in the kitchen, where it single-handedly takes on the role that a béchamel would normally play. It is a cheese that should be purchased and consumed as close as possible to the time it was made — giving this lasagna a delectable flavor.
Put the flour in a medium-sized bowl and create a well in the center. Add the eggs and egg yolks and beat them with a fork, incorporating a little flour from the sides of the mound at a time, until a sticky mass is formed.
Turn the dough out onto a lightly floured work surface and knead the dough until smooth and pliable. Wrap the dough in plastic wrap and let rest at room temperature for 1 hour.
Trim the tips of the asparagus and set aside. Bring a pot of salted water to boil over high heat. Blanch the spears of asparagus for about 8-10 minutes, or until very tender. Remove from the water, shock in a bowl of ice water, and drain. Add the reserved asparagus tips to the boiling water and cook for about 4-5 minutes, or until cooked but still firm. Shock the tips and drain; reserve the cooking water for cooking the pasta.
In the bowl of a food processor fitted with the blade attachment, purée the cooked asparagus spears (not the tips) until smooth. Add the burrata and the Parmigiano-Reggiano. Purée until smooth. Season with salt and pepper, to taste.
When the dough is rested, roll out with a pasta machine to a thickness of 1/32-inch. Cut into sheets roughly 6-by-12 inches. Boil the pasta sheets in the same water the asparagus was cooked in for about 1 minute, or until al dente. Shock the pasta in ice water quickly and then remove and pat dry with a clean kitchen towel.
Preheat the oven to 425 degrees.
Spoon ¼ cup of the asparagus-burrata filling into the bottom of a lightly buttered 8-by-12-inch baking dish. Place a sheet of cooked pasta over the filling and then cover with another ¼ cup of the filling. Scatter a few of the cooked asparagus tips over the filling and then sprinkle with a bit of the Parmigiano and a drizzle of the olive oil. Place another cooked sheet of pasta on top and continue layering, ending with the last of the sauce. Tear the burrata into golf ball-size pieces and scatter over the top. Sprinkle with the remaining Parmigiano and the last of the olive oil.
Place on the top rack of the oven and bake for 20-25 minutes, or until hot and bubbling with golden melted pools of burrata and crisp edges of pasta.
Remove from the oven and let stand for 10 minutes before serving. Cut into squares and place individual pieces on warm dinner plates. If using the prosciutto, gently drape a few slices around the edges of the lasagna and serve.