1. Preheat oven to 350°F (175°C). Grease and flour a 15×10-inch jelly roll pan or line with parchment paper.
2. Sift together flour, cocoa powder, baking powder, and salt; set aside.
3. In a large bowl, beat egg yolks with 1/2 cup granulated sugar until thick and pale. Stir in vanilla bean paste.
4. In a separate clean bowl, beat egg whites until soft peaks form. Gradually add the remaining 1/4 cup granulated sugar and beat until stiff peaks form.
5. Fold dry ingredients into the yolk mixture, then fold in egg whites gently.
6. Pour batter into the prepared pan, spread evenly, and bake for 10-12 minutes.
7. Invert the warm cake onto a towel dusted with powdered sugar. Peel off parchment paper. Roll cake and towel together and let cool.
8. For buttercream: beat softened butter until fluffy. Gradually add powdered sugar and cocoa powder. Beat in milk, vanilla, and salt.
9. Unroll cooled cake, spread with 1 cup of buttercream, and re-roll. Trim and attach a piece for a 'branch' with frosting.
10. Frost with remaining buttercream, create bark patterns, dust with powdered sugar, and decorate as desired.