1. Process heirloom tomatoes in a food processor 30 to 60 seconds or until pureed. Line a colander with 2 layers of cheesecloth; place colander over a large bowl. Pour tomato puree into cheesecloth. Tie ends of cheesecloth together. Chill 24 hours to allow puree to drain.
2. Remove bowl from refrigerator, and gently press cheesecloth using back of a spoon to extract remaining liquid. (Yield should be about 4 cups.) Cover and chill.
3. Preheat oven to 250°. Core medium tomatoes, removing stem ends. Cut tomatoes in half; place, cut sides up, on a lightly greased aluminum foil-lined baking sheet; sprinkle with 1 Tbsp. olive oil, 1/4 tsp. sea salt, and 1/4 tsp. sugar. Bake 2 hours.
4. Place 1 roasted tomato half in each of 4 serving bowls. Top with squash and next 4 ingredients; sprinkle with thyme and 1/4 tsp. salt. Add remaining 1/4 tsp. salt to tomato water, and divide tomato water among bowls. Drizzle with remaining 1 Tbsp. olive oil, and serve immediately.