Step 1: Preheat oven to 275-degrees F. Cut out the hard core of each of the tomatoes and make an X in the smooth flower end. Plunge into boiling water for about 15 seconds, or until the skins loosen, then plunge into ice water; drain. Peel; cut plum tomatoes in half, regular tomatoes into quarters, then remove all the seeds and pulp.
Step 2: Cover a baking sheet with aluminum foil and brush with the olive oil. Place the tomatoes cut side down on the pan. Scatter the garlic and thyme around and sprinkle with the salt.
Step 3: Bake for 2 hours or more, turning the tomatoes every 30 minutes or so to make sure that they are not browning (if they are, lower the heat) and turning the baking sheet so the tomatoes cook evenly. The tomatoes are done when they are very soft and shriveled. They will keep, refrigerated, for a few days.
Step 4: Make a bed of arugula on each of the four plates. Slice the squashes into long think ribbons (a mandolin is ideal for this.) Toss them with the olive oil and lemon juice, then lay the strips on the arugula; pour any remaining dressing over them.
Step 5: Garnish with the Parmesan and tomato confit; drizzle with a little of the confit's oil if you have it. Sprinkle salt and pepper and serve.
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