Place white chocolate chips and 3/4 cup heavy cream in a medium microwave-safe glass or ceramic bowl. Microwave at 50% power until chocolate is completely melted and cream is fully incorporated, about 2 minutes, stopping and stirring every 30 seconds. Cool to room temperature, 10 to 20 minutes, then refrigerate until completely chilled, at least 4 hours to overnight.

Meanwhile, place dark chocolate chips and 3/4 cup heavy cream in a separate medium microwave-safe glass or ceramic bowl. Microwave at 50% power until chocolate is melted and cream is fully incorporated, about 2 minutes, stopping and stirring every 30 seconds. Cool to room temperature, 10 to 20 minutes, then refrigerate until completely chilled, at least 4 hours, to overnight.

Preheat the oven to 350 degrees F (175 degrees C). Line a 12x17-inch baking sheet with parchment paper; grease baking sheet and parchment paper.

Beat egg yolks and brown sugar in a large bowl with an electric mixer on medium-high speed until light and creamy, about 2 minutes. Beat in butter, coffee, and vanilla extract.

Sift flour, 1/3 cup cocoa powder, baking powder, baking soda, and salt together in a small bowl. Add to egg yolk mixture; mix until just combined.

Beat egg whites in a separate clean, dry, large glass, metal, or ceramic bowl with an electric mixer with clean, dry beaters on high speed until soft peaks form. Gradually add sugar, 1 tablespoon at a time, continuing to beat until glossy and stiff peaks form. Gently fold 1/3 egg white mixture into yolk mixture until just combined. Gently fold in remaining egg white mixture until no egg white streaks remain. Do not overmix.

Pour batter into the prepared baking sheet; gently spread evenly.

Bake in the preheated oven until cake springs back lightly when touched, about 10 minutes. Do not overbake.

While cake is baking, lightly sprinkle a piece of parchment paper slightly larger than the baking sheet with remaining 1 tablespoon cocoa powder.

Run a table knife around edges of baking sheet to loosen cake. Invert quickly and carefully, the baking sheet will be hot, onto the prepared parchment paper. Remove and discard parchment paper the cake was baked on. Working quickly, roll cake and cocoa-sprinkled parchment paper together, starting from a narrow end. Set cake, seam-side down, on a wire rack to cool completely, 1 to 2 hours.

While cake cools, whip white chocolate (it will be soft-set) with an electric mixer, starting on low speed and working up to medium-high, until light and fluffy, 1 to 2 minutes. Add 3/4 cup heavy cream; beat on medium-high speed until reaches texture of whipped cream, 1 to 2 minutes more. Spoon white chocolate whipped cream into a piping bag or a resealable plastic bag with a 1/2-inch hole cut into a corner. Refrigerate filling until ready to use.

Whip dark chocolate (it will be firm set) with an electric mixer, starting on low speed and working up to medium speed, until it crumbles and becomes lighter in color and texture, 1 to 2 minutes. Add remaining 3/4 cup heavy cream; beat on medium-high speed until smooth, light, and fluffy, 1 to 2 minutes (dark chocolate mixture will be slightly thicker than white chocolate mixture). Add more heavy cream if necessary to achieve desired "lightness" and consistency of whipped mousse. Spoon whipped dark chocolate mousse into a piping bag or a resealable plastic bag with a 1/2-inch hole cut into a corner. Refrigerate filling until ready to use.

Gently unroll cooled cake; discard parchment paper. Pipe white and dark chocolate fillings in alternating vertical stripes (about 1/2-inch thick) along 12-inch side of cake until entirely covered, leaving about 1/2-inch border on all sides. Gently reroll cake over fillings (without parchment) as tightly as possible without breaking the cake until completely rolled up. Place on a serving platter. Refrigerate until set, at least 1 hour.