1. Prepare Rabbit: Preheat oven to 300°. Bring dried mushrooms and 1/2 cup chicken broth to a boil in a small saucepan over medium heat. Remove from heat, and let stand 30 minutes.
2. Meanwhile, sprinkle rabbit with kosher salt and freshly ground pepper. Cook rabbit, in 2 batches, in hot oil in a heavy ovenproof skillet or enameled cast-iron Dutch oven over medium heat 8 to 10 minutes on each side or until deep golden brown. Transfer to a wire rack in a jelly-roll pan, reserving drippings in skillet.
3. Add carrots and garlic to reserved drippings in skillet, and sauté 4 minutes or until carrots are softened. Stir in wine, port, mushroom mixture, and remaining 2 cups broth; bring to a simmer, stirring occasionally. Skim off foam and fat.
4. Remove and discard root end of leek; remove dark green top of leek. Tie together green leek top, parsley sprigs, and next 2 ingredients with kitchen string. Add herb bundle and rabbit to skillet; bring to a simmer, and cover.
5. Braise at 300° for 15 minutes. Transfer rabbit loin pieces to wire rack in jelly-roll pan. Braise remaining rabbit 30 more minutes. Transfer remaining rabbit to wire rack; cool 15 minutes.
6. Meanwhile, bring liquid in skillet to a simmer over medium heat, and cook 15 minutes or until liquid is reduced by half. Skim off foam and fat. Pour through a fine wire-mesh strainer into a small saucepan; discard solids. Cover and keep warm over low heat.
7. Remove rabbit from bones; shred with 2 forks. Discard bones.
8. Prepare Pilau: Cut reserved leek in half lengthwise; rinse thoroughly under cold running water to remove grit and sand. Cut leek into 1/2-inch-thick slices.
9. Melt 2 Tbsp. butter in a Dutch oven over medium-high heat. Add carrots, next 2 ingredients, and leek; sauté 2 minutes or until onions and leek just begin to soften. Add rice, and cook, stirring constantly, 1 minute. Add broth, and bring to a boil. Cover, reduce heat to low, and cook 15 minutes or until rice is tender and broth has been absorbed.
10. Stir rabbit into rice; transfer to a platter. Whisk remaining 2 Tbsp. butter into warm braising liquid. Spoon liquid over pilau before serving.