Preheat the oven to 375 °F. Lightly grease an eight-cup casserole dish that fits a single lasagna sheet nicely.
In a large sauté pan over medium heat add the oil and onion and celery and sauté until the onion is translucent, about five minutes. Add the garlic and sauté one minute more. Add the chicken stock and then stir in the squash and sage and bring to a simmer, cooking loosely covered until the squash is tender (about 12 minutes for fresh, and six minutes for frozen. Stir in the chicken and sundried tomatoes. Remove this from the heat and stir in the ricotta and milk. Season to taste.
To assemble, spoon a little of the chicken filling at the bottom of the prepared dish and place a sheet of lasagna on top. Spoon a generous layer of the chicken filling over this and top with a sprinkling Asiago. Top this with a lasagna sheet and repeat with the chicken filling and Asiago three more times, ending with a generous layer of Asiago on top. Place a sheet of parchment over the lasagna and cover with foil. Bake covered for 30 minutes, then uncover and bake an additional 35 to 45 minutes, until the top is golden and melted and the filling is bubbling around the sides. Let the lasagna cool for ten minutes before scooping or slicing to serve.
Additionally the lasagna can be assembled a day ahead and chilled, but add another 15 to 20 minutes of baking while it is covered.