Heat the oil in a deep 12-inch or larger cast iron or enamel-coated cast iron skillet over medium heat until shimmering.
Add the diced onions and cook until translucent and soft, about 5 minutes.
Increase the heat to medium-high. Remove the sausage from its casing and add to the skillet, breaking the meat into small pieces with a wooden spoon, until the sausage is cooked through, about 5-7 minutes.
Add the mushrooms and sauté until tender, about 5 minutes.
Add the marinara and broth, stir to combine, and bring to a simmer.
Using your hands, break the dried lasagna noodles into 1 1/2- to 2-inch pieces and add to the skillet. Stir to combine and spread into an even layer, making sure as many of the noodles are submerged in the sauce as possible.
Reduce the heat to a simmer (small bubbles all over), cook uncovered, stirring occasionally and watching if noodles are sticking to the bottom. If they are, lower the heat as needed.
Note: Add 1/4 cup at a time of water or extra broth as needed if the noodles have absorbed all the liquid before they're cooked. Make sure the noodles are submerged in the sauce.
Cook until the noodles are just tender, and the sauce has thickened slightly, about 25 minutes. Taste and add a pinch or two of sea salt if needed.
When the noodles are cooked through, turn the heat off the skillet and add dollops of the ricotta cheese on top of the lasagna with a small spoon. Flatten out the ricotta dollops with the back of the spoon, this will help it to melt quicker. Do not stir anymore.
Remove the pan from the heat and sprinkle withthe mozzarella. Cover and let sit until the cheeses are melted, about 5 minutes. (If you like a browned top, place the ovenproof skillet under the broiler for a few minutes until the cheese is golden-brown in spots.)
Sprinkle with optional chopped parsley or basil leaves and serve.