Bring the pork in barbecue sauce to a simmer in a large pot over medium heat; keep warm at a slow simmer.
Toss the mozzarella cheese and Cheddar cheese together in a bowl. Beat the ricotta cheese, egg, salt and pepper together in a separate bowl until smooth.
Spread a layer of the barbecue sauce into the bottom of a large, deep baking dish. Lay enough lasagne sheets over the sauce to cover the bottom of the dish.
Spread an even layer of the pulled pork in barbecue sauce over the lasagne. Top with a layer of the ricotta mixture and the cheese mixture.
Repeat layering. Finish the assembly by topping with a layer of the ricotta mixture, a thin layer of the pork and a light sprinkling of the cheese mixture. Reserve a small portion of the cheese mixture for later.
Bake in the preheated oven for 40 minutes. Top the lasagne with the reserved cheese mixture and return to over until the cheese is bubbly, about 5 minutes. Remove from oven and allow to rest for 15 minutes before serving.