Place white chocolate chips and 3/4 cup heavy cream for white chocolate filling in a medium microwave-safe bowl. Microwave on 50% power until chocolate is completely melted and cream is fully incorporated, about 2 minutes, stopping to stir every 30 seconds. Set aside and allow to cool to room temperature, 10 to 20 minutes. Place into the refrigerator until completely chilled, at least 4 hours, to overnight.

Meanwhile, place dark chocolate chips and 3/4 cup heavy cream for dark chocolate filling into another medium microwave-safe bowl. Microwave on 50% power until chocolate is melted and cream is fully incorporated, about 2 minutes, stopping to stir every 30 seconds. Set aside and allow to cool to room temperature, 10 to 20 minutes. Place into the refrigerator 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 the pan and parchment.

Combine egg yolks and brown sugar in a large bowl. Beat at medium-high speed until mixture is light and creamy, about 2 minutes. Mix in butter, coffee, and vanilla extract.

Sift flour, 1/3 cup cocoa, 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 bowl with clean, dry beaters on high speed until soft peaks form. Gradually add sugar, 1 tablespoon at a time, beating constantly. Continue to beat until mixture is glossy and stiff peaks form. Fold 1/3 of the egg white mixture gently into the yolk mixture until just combined. Fold in remaining egg whites gently until no streaks of egg white remain. Do not overmix.

Pour batter into the prepared pan and 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, prepare a piece of parchment paper that is slightly larger than the baking pan. Sprinkle lightly with remaining cocoa powder.

Remove cake from the oven and run a knife around the edges of the pan to loosen the cake. Invert the cake quickly and carefully, as pan will be hot, onto the cocoa-sprinkled parchment paper. Remove the piece of parchment the cake baked on and discard. Working quickly, roll the cake and cocoa-sprinkled paper up together, starting from the narrow end. Set the cake seam-side down on a wire rack to cool completely, 1 to 2 hours.

While cake cools, finish the fillings. Remove the white and dark chocolate from the refrigerator. The white chocolate will be soft-set. Whip the white chocolate, beginning on low speed and working up to medium-high, until light and fluffy, 1 to 2 minutes. Pour in remaining 3/4 cup of heavy cream, and beat on medium-high speed until mixture has the texture of whipped cream, 1 to 2 minutes. Place white chocolate whipped cream into a piping bag or zip-top bag with a 1/2-inch hole cut into a corner. Keep refrigerated until ready to use.

Prepare dark chocolate filling. The dark chocolate will be firm set. Whip the dark chocolate, starting on low speed and working up to medium speed, until mixture crumbles. Beat until the chocolate crumbles begin to become lighter in color and texture, 1 to 2 minutes. Add remaining 3/4 cup heavy cream and beat on medium-high speed until mixture is smooth, light, and fluffy, 1 to 2 minutes. The dark chocolate mixture will be slightly thicker than the white chocolate. Add more heavy cream if necessary to achieve the desired "lightness" and consistency of whipped mousse. Place whipped dark chocolate mousse into a piping bag or zip-top bag with a 1/2-inch hole cut into a corner. Keep refrigerated until ready to use.

Unroll cooled cake gently and discard parchment paper. Pipe alternating stripes of white chocolate whipped cream and whipped dark chocolate mousse vertically along the 12-inch side of the cake, in stripes about 1/2-inch thick. Continue this vertical pattern along the entirety of the cake, leaving about a 1/2-inch border on all sides. Gently roll the cake back up over the filling (without the parchment) as tightly as possible without breaking the cake until the cake is completely rolled up. Place onto a serving platter and refrigerate until set, at least 1 hour.