Heat oil in a large skillet over medium-high heat. Cook mushrooms and onion for 8 minutes until the released liquid evaporates, then add beef and cook for 7 to 8 minutes, breaking it into small crumbles, until 160°F; add garlic for 1 minute.
Transfer to a 6-quart slow cooker and stir in passata, the 2 1/2 cups broth, basil, salt and pepper. Fold in dry ziti until every tube is coated and settled into the liquid.
Cover and cook on low for 2 to 2 1/2 hours, stirring gently after 1 hour. Begin checking the center pasta at 90 minutes and stop when tender, adding some of the measured extra broth only if the tubes are still firm and the sauce is nearly absorbed.
Drop ricotta in small spoonfuls over the hot pasta, sprinkle with mozzarella and Parmesan and cover for 10 to 15 minutes until the cheese melts and the center is 165°F.
Switch off and rest uncovered for 10 minutes. Spoon into six bowls without fully stirring away the ricotta pockets; the ziti goes into the cooker dry and needs no separate boiling.