Place rack in upper-third and preheat oven to 375°F.
Meanwhile, peel 1 honeynut squash and cut in half length-wise. Scoop out any seeds and discard. Thinly slice remaining halves, going as thin as you can without being too fussy. Dice 1 shallot and 4 garlic cloves. Set aside.
Heat a large braiser, or cast iron, on medium-high heat and add 2 tbsp olive oil. Tear the ground pork into large pieces and place in the hot pan. Cook undisturbed for 5 minutes, or until golden brown on one side.
Meanwhile, tear the thyme leaves from the stems and roughly chop. Pile 10 sage leaves on top of each other and roll them into a tube. Thinly slice the tube, leaving you with long strands. Roughly chop the strands and set aside.
Back in the pot, use a wooden spoon, or spatula, to break the large pieces into small bits, they should be the size of a dime or smaller. Season generously with 2 tsp salt. Give it a mix and let cook, occasionally stirring, until at least 1/3 of the pork is deep golden brown. Meanwhile, grate your cheese, if you haven't already.
When the pork is brown, add in your diced shallots and garlic. Turn heat to medium-low and cook until the shallots have softened and the garlic is very fragrant. Stir in chopped thyme and sage, and 1 tbsp tomato paste. Cook until the tomato paste begins to caramelize on the bottom of the pan, about 2 minutes.
Sprinkle 1/4 cup all purpose flour over the seasoned pork and mix. Let it cook for about a minute, or until the raw flour smell is gone. Then, slowly whisk in 2 cups milk, followed by 2 cups broth. It will be thick and paste-like at first, then gradually become a soup-y liquid. Taste and adjust seasoning to your preference.
Break 8 lasagna sheets into smaller pieces. Add 1/2 of the broken lasagna sheets to the pan in an even layer. Then, spread out 1/2 of the sliced squash. Top with 1/2 the grated fontina. Repeat all three layers once more, then use your spoon, or spatula to press the layers into the broth, taking an extra moment to stir up the pork that might be stuck on the bottom — this will give it a chance to float in between the layers and spread evenly throughout. Top with 1/2 cup panko bread crumbs and 1/2 cup grated parmesan. Scatter the remaining sage leaves on top and cover each in a drizzle of olive oil. Bake at 375° for 35 minutes.
Switch oven to broil and cook until the sides are bubbling and the top is golden brown and crispy, about 5 minutes. Remove and let sit for another 5 minutes before serving.
