Prep the loaf pan. Lightly grease a 10 x 5 loaf pan (1.5 lbs) and set it on a foil-lined baking sheet. Set aside. Remove the burrata from their containers and lay them out on paper towels to drain off any excess water. Refrigerate until ready to assemble.
Make the vodka sauce. Heat the olive oil in a medium saucepan over medium-low heat. Add the diced pancetta and cook, stirring occasionally, until the pancetta is crispy. Use a slotted spoon to remove the crispy pancetta from the pan and transfer to a small bowl. Add the shallot and garlic to the pan and cook for 2-3 minutes, until the shallots are soft. Then, add in the tomato paste, oregano, and red pepper flakes and cook for another 2-3 minutes, stirring often, until the tomato paste has darkened. Add the crushed tomatoes, crispy pancetta, and sugar and bring to a simmer for 15 minutes, until the sauce has reduced slightly. Stir in the vodka off the heat then add the cream and simmer for 10 more minutes. Season to taste with salt and pepper.
Make the bechamel. While the vodka sauce simmers, melt the butter in a small saucepan over medium heat. Once the butter is melted and foamy, add the flour and whisk to form a paste. Continue to whisk while cooking for another 2 minutes, to cook out the raw flour flavor. While whisking, stream in the warmed whole milk with the nutmeg. Continue to whisk while the sauce simmers for 4-6 minutes, until very thick. Remove from the heat, stir in the pesto, and season to taste with salt and pepper.
Preheat an oven to 350 F.
Layer the lasagna. Ladle a thin layer of vodka sauce into the prepared loaf pan. Add a layer of lasagna noodles. Break them as needed so that they fit in the pan and don’t significantly overlap. Spread 3/4 cup of the vodka sauce over the noodles and add about 1/3 of the mozzarella over the top, then top with more noodles. Spread 1/2 of the bechamel over the noodles, then tear apart a piece of burrata and lay it over the bechamel. Sprinkle 1/3 of the parmesan cheese over the top. Repeat each of the layers one more time. Spread the remaining vodka sauce over the top of the final layer of noodles and wrap with foil. Reserve the remaining mozzarella and parmesan for later.
Bake. Bake covered for 40 minutes. Remove the foil and top with the mozzarella and parmesan and bake for another 20 minutes, until the lasagna is bubbling and the cheese is melted. Let the lasagna cool for 30 minutes before serving.