Heat oil in a skillet over medium-high heat and cook mushrooms for 7 to 9 minutes until their liquid evaporates. Add garlic for 1 minute, remove from heat and cool for 5 minutes.
Blend cottage cheese, egg, Parmesan, 1/4 teaspoon salt and pepper until smooth. Fold in dried spinach and cooled mushrooms, then spoon into a sturdy piping bag or food-storage bag with one corner snipped.
Fill each dry jumbo shell with an equal share of the mixture, about 2 tablespoons, supporting the shell gently so it doesn’t crack. Do not boil or soak the shells first.
Mix passata, broth, basil and remaining salt. Butter a 6-quart slow cooker, spread 1 cup sauce over the bottom and arrange twelve filled shells, then add half the remaining sauce.
Add the other twelve shells, cover completely with remaining sauce and cover the cooker. Cook on low for 3 1/2 to 4 hours, checking after 3 hours, until a center shell is tender at its folded edge and the filling reaches 165°F.
Scatter mozzarella over the hot shells, cover for 10 minutes to melt, then switch off and rest uncovered for 10 minutes. Serve four shells per person with sauce.