Step 1
If using fresh pasta, cut 8 oz. fresh lasagna sheets crosswise into sixths, then in half once or twice to create a rough square shape; set aside. If using dried pasta, snap 9 oz. dried lasagna noodles (about 11) into pieces of a similar size and shape; set aside.
Step 2
Place half (about 5 cups) of 1 small head of green cabbage (1½–2 lb.), thinly sliced, in a medium bowl and sprinkle 1 Tbsp. Diamond Crystal or 1¾ tsp. Morton kosher salt over. Massage salt into cabbage with your hands until cabbage begins to soften and wilt, about 4 minutes, depending on how aggressive your massage techniques are. Set aside.
Step 3
Toast 1 tsp. fennel seeds and 1 tsp. mustard seeds in a small saucepan over medium-high heat, stirring often, until fragrant, about 3 minutes. Add 5 garlic cloves, coarsely chopped, and 1 cup heavy cream and bring to a simmer. Immediately remove from heat and let sit until cream is infused with flavor, 5–7 minutes. Strain through a fine-mesh sieve into a small bowl; discard solids. Set infused cream aside.
Step 4
Heat 2 Tbsp. extra-virgin olive oil in a large skillet or a medium Dutch oven over medium-high. Cook one 13-oz. link kielbasa, halved lengthwise, sliced into ¼”-thick half-moons, and cook until browned, 5–8 minutes. Using a slotted spoon, transfer to another small bowl, leaving fat in pan.
Step 5
Arrange 1 lb. oyster mushrooms, torn into large irregular pieces, in pan in a single layer and cook, undisturbed, until mushrooms start to take on a bit of color underneath, 7–9 minutes. Season with a pinch of salt and freshly ground pepper and continue to cook, turning and pressing occasionally, until golden brown all over, 5–8 minutes more. Transfer mushrooms to bowl with kielbasa.
Step 6
Place remaining cabbage (about 5 cups) in same pan and season with a pinch salt; drizzle in more oil if pan looks dry. Cook, stirring occasionally, until cabbage is wilted and decreased in volume and browned around some of the edges, 12–15 minutes. Add reserved infused cream, half (about ½ cup) of 1 bunch dill, coarsely chopped, and reserved salted cabbage. Cook, stirring often, until combined and liquid reduces slightly, about 5 minutes. Remove from heat and return kielbasa and mushrooms to pan.
Step 7
Meanwhile, cook pasta squares in a large pot of boiling salted water until al dente, about 3 minutes if using fresh or according to package directions if using dried. Drain, reserving 1 cup pasta cooking liquid.
Step 8
Transfer pasta to pan with kielbasa and mushrooms and toss, adding pasta cooking liquid a splash at a time, until combined and a glossy, thick sauce forms. Taste and season with more salt and pepper if needed.
Step 9
Divide lazanki among shallow bowls and top each with a dollop of sour cream (if using) and remaining dill.