Cook the vegetables slowly in a flavorful liquid for our interpretation of leeks vinaigrette. The leek halves will slightly overlap in the pan. Pancetta is Italian cured pork. If you can't find it, substitute your favorite cured bacon.
Preheat oven to 325°.
To prepare leeks, arrange leek halves in an 8-inch baking dish; sprinkle evenly with 1/4 teaspoon salt and 1/4 teaspoon pepper. Add broth and next 3 ingredients (through 1 thyme sprig). Cut 1 (8-inch) square of parchment paper; lightly coat with cooking spray. Place parchment over leek mixture, coated side down. Bake at 325° for 50 minutes or until leeks are tender. Let stand 5 minutes; drain cooking liquid through a sieve over a bowl, reserving solids.
Place cooking liquid in a small, heavy saucepan; bring to a boil. Cook until reduced to 1/4 cup (about 8 minutes). Chop cooked carrot; set aside. Coarsely chop cooked garlic; set aside.
To prepare dressing, heat oil in a small skillet over medium-high heat. Add pancetta to pan; sauté 5 minutes or until crisp. Stir in garlic and chopped leek; sauté 2 minutes, stirring occasionally. Sprinkle with sugar; sauté 1 minute or until sugar dissolves. Stir in vinegar; simmer 2 minutes. Add braising liquid, 1/8 teaspoon pepper, and dash of salt; simmer 2 minutes or until slightly thick. Remove from heat. Arrange leek halves in a serving dish; sprinkle with carrot. Drizzle pancetta mixture over leek halves. Garnish with thyme sprigs, if desired.