Cut squash into ⅜-inch-thick slices, coat with 1 tbsp olive oil and set aside. Heat 1 tsp oil in a large skillet over medium-high heat; sauté mushrooms, ½ tsp salt and pepper until golden and soft, 5 min. Transfer to a bowl and stir in onion and spinach, set mixture aside.
In the same skillet cook squash, turning once, until golden and just tender, about 8 min. In a food processor blend goat cheese, ricotta, vinegar, sugar and ½ tsp salt until smooth (alternately blend well with a hand mixer). Spread half the cheese mixture into the bottom of each tart shell. Top with mushroom mixture and squash slices. Serve warm or at room temperature. Can be chilled up to 6 hr; bring to room temperature before serving.
Here's a trick for removing the rim. Set the tart on top of an overturned bowl. The rim can be gently nudged off and will come off, leaving your tart on it's base. Lift the tart off.For rectangle tarts like this one, use two cans. Two little tomatoe paste cans work like a charm, but other food cans will do as well!