Preheat oven to 350°F. Grease bottom and sides of a 12-by-15-inch jelly-roll pan and line bottom with parchment. Pour prepared brownie batter into pan; bake until just done, 12 to 15 minutes. Remove pan to a wire rack and let brownie cool.
Once brownie has cooled, run a knife along sides of pan to loosen it. Place a piece of parchment as big (or larger) as jelly-roll pan onto a sheet pan. Invert brownie onto sheet pan; peel off parchment.
Refrigerate ice cream to soften, 10 to 20 minutes. In a medium bowl, mix honey, vanilla and peanut butter. Using a knife or offset spatula, carefully spread peanut butter mixture evenly over brownie, leaving a 1-inch border on each long side.
Using a spoon, scoop out thin slabs of softened ice cream and arrange along center of peanut butter layer. Spread ice cream just down middle of brownie.
With one long side of pan facing you, carefully lift parchment under brownie and roll brownie into a jelly roll, laying it seam side down to seal. Put in freezer immediately and freeze for at least 2 hours. Dust with unsweetened cocoa powder, if desired.