Heat the oven to 350 °F and butter a 9x13 casserole dish. The dish should be fully coated so the pasta mixture settles in easily and serves cleanly later.
Boil the linguine in a large pot of well-salted water until barely al dente, then reserve 1/2 cup of the cooking water, drain, and keep the pasta loosely covered. The noodles should still have a slight bite because they will keep softening in the oven.
Set a large skillet over medium-high heat, add the olive oil, and cook the mushrooms for about 3 min until they soften before adding the onion for 5 to 7 min and the garlic for 1 more min. The mushrooms should release their moisture, the onions should turn soft and lightly golden, and the garlic should smell fragrant but not browned.
Move the cooked vegetables to the board with the chicken, then use the same pot to melt the butter and whisk in the flour for about 1 1/2 min. The roux should turn lightly golden and smell nutty, which tells you the flour has cooked enough for a smooth sauce.
Slowly whisk in the chicken broth, lemon juice, salt, and pepper until smooth, then add the half & half and bring the sauce to a gentle simmer. The sauce should look creamy and lump-free, and it may seem a bit thin at first, which is fine because it thickens more as the pasta bakes.
Return the pasta to the pot and stir in the chicken, mushroom mixture, parsley, and all of the sauce, adding up to 1/2 cup of reserved pasta water if you want a looser mixture. Everything should look generously coated, with enough sauce to settle around the noodles instead of disappearing into them.
Spread the mixture evenly in the prepared casserole dish and scatter the mozzarella over the top. The surface should be level so the cheese melts evenly and the center heats through at the same pace as the edges.
Cover with foil and bake for 20 min, then uncover and bake 15 min more, broiling for 1 to 2 min at the end if you want golden spots on top. The casserole is ready when it is hot and bubbly, the cheese has melted fully, and the top has light browning in places.