Heat a small saucepan over medium-high heat and add the oil, onions, shallots, and a heavy pinch of salt; toss to coat with oil. Reduce heat to low and allow onions and shallots to soften, then caramelize for about 15 minutes, stirring often. Chop tomatoes coarsely. Combine all ingredients in a large saucepan and bring to a boil, uncovered. The pan should be no more than half full, because jam tends to bubble up so much that it might overflow. Cook about 30–40 minutes, stirring often, until reduced and thickened. The jam takes on a translucent quality when it is getting close to done. To test, put a drop on a chilled plate—if it holds together, it’s done; if it leaks liquid at the edges, it needs a little more time.

Combine flour, pine nuts, salt, baking powder, and pepper in a food processor; pulse 3 times or until combined. Combine oil and 3 tablespoons ice water in a small bowl. With processor on, slowly add oil mixture through food chute, and process until dough is crumbly. Sprinkle dough into a 9-inch glass or ceramic pie plate coated with cooking spray. Press dough into an even layer in bottom and sides of dish. Bake for 10 minutes. Remove from oven. Make a day ahead, if desired. Cover and store in refrigerator.

Preheat the oven to 400°F.

Spread the Tomato-Onion Jam evenly over the prepared tart crust. Layer the tomatoes, eggplant, yellow squash, and zucchini in overlapping rows, or as desired. Sprinkle the roasted red peppers around the tart. Drizzle the assembled tart with a little olive oil, making sure to brush a little on the edges; then season with salt and pepper, and top with half of the fresh oregano. Bake for 25 minutes uncovered, then cover with foil and cook for 20 minutes longer. Remove from the oven and immediately sprinkle with the goat cheese and remaining oregano before serving.