Peel shrimp, reserving shells and tails. Cut shrimp crosswise into thirds; set aside. Place shells and tails in a medium saucepan. Add 1 tablespoon oil, carrots, celery, onion, and minced garlic. Cook over medium-high heat for 1 minute. Add 1 cup of wine, and cook until reduced to about 1/3 cup. Strain through a fine sieve; discarding solids.
Bring a large pot of water to a boil. Add salt and spaghetti, and cook until al dente, according to package directions.
In a large straight-sided skillet, heat remaining 1/4 cup oil over medium-high heat. Add sliced garlic, and cook, stirring occasionally, until just beginning to turn golden brown. Add shrimp, and remaining 1/2 cup wine. Cook, stirring, until almost cooked through. Add strained cooking liquid, and cook until shrimp are just firm. Stir in butter and parsley.
Drain spaghetti, and place on a serving platter. Top with shrimp mixture, and serve immediately.