Cake: Coat two 9 x 2-inch round baking pans with nonstick cooking spray. Line bottoms with waxed paper; spray paper.
Break up chocolate into a small microwave-safe bowl; add oil. Microwave on HIGH power for 1 minute; stir until smooth. If needed, heat an additional 15 seconds. Cool slightly.
In medium-size bowl, mix flour, cocoa, baking soda and salt. In large bowl, with mixer on medium speed, beat butter and sugar until fluffy, about 3 minutes. Add eggs, one at a time, beating after each. Beat in chocolate mixture and vanilla.
Place vinegar in a 2-cup capacity measuring cup. Add milk and stir until blended. On low speed, beat flour mixture into butter mixture, alternating with milk, beginning and ending with flour; beat 1 minute. Spread into pans.
Bake at 350°F for 35 minutes or until toothpick inserted in centers tests clean. Cool cakes in pans on wire racks 15 minutes. Turn cakes out onto racks; cool layers completely (about 2 hours).
Prepare Frosting: In large glass bowl, microwave white chocolate on 50% power for 1 minute; stir. Microwave another 30 seconds until melted and smooth. Let cool slightly.
With mixer, beat in sour cream and butter until smooth. Refrigerate 10 minutes. On low speed, beat in sugar until blended; continue to beat until fluffy. If too soft, refrigerate 10 minutes. Spread 1 cup frosting on top of one layer. Stack with second cake layer. Spread thin coating of frosting around side of cake. Refrigerate or freeze 15 minutes, until firm. Frost side and top with remaining frosting; garnish with curls, if desired. Refrigerate until set, 20 minutes.