Make the cake
Position a rack in the bottom third of the oven and heat the oven to 350°F.
Butter an 18x13-inch rimmed baking sheet. Line the pan with parchment. Butter the parchment and sprinkle the 2 Tbs. of flour over it, shaking the pan for even coverage and knocking out the excess.
Put the carrots in a 1-quart saucepan and cover with water. Bring to a simmer over medium heat and cook until tender, 20 to 30 minutes. Drain well. Purée in a food processor until smooth. Transfer to a small bowl, add the orange zest, and cool to room temperature
In a small bowl, sift together the 3/4 cup flour, cinnamon, baking powder, baking soda, nutmeg, ginger, pepper, and cloves. Add the salt and stir to combine.
In a stand mixer fitted with the whisk attachment, beat the egg yolks, brown sugar, and granulated sugar on medium-low speed until combined. Increase the speed to medium high and beat until light and fluffy, 3 to 4 minutes. Add the carrot purée and mix on low speed, scraping down the sides as necessary, until just combined. In three additions, add the flour mixture, mixing on low speed, and scraping the bowl as needed, until just combined. Transfer the mixture to a large bowl.
In a clean stand mixer bowl, with a clean whisk, beat the egg whites on medium-high speed until medium peaks form, about 2 minutes. Fold one-third of the beaten whites into the carrot mixture to lighten it. Gently fold in the remaining whites until the batter is evenly colored, with no streaks of white. Transfer the batter to the prepared pan, gently spreading it with an offset spatula.
Bake, rotating the pan about halfway through, until golden-brown, set to the touch, and beginning to pull away from the sides of the pan, 16 to 18 minutes. Immediately loosen the cake from the sides of the pan with a paring knife. Sift the confectioners’ sugar evenly over the cake.
While the cake is still hot, lay a clean, unscented, lint-free kitchen towel over it. If the towel is long, extend one end about 1-1/2 inches beyond a short side of the cake and don’t worry about the other end. Invert a large rack or cutting board over the towel. Holding both the baking sheet and the rack with protected hands, invert the cake. Remove the baking sheet and parchment.
Using both hands and starting from the short end with the shortest towel overhang, roll the cake and the towel up together. Let cool for 30 minutes.
Carefully unroll the cake; it should look wavy and both ends should curl. (It’s OK if there are some small cracks.) Let cool completely, 10 to 15 minutes.
Using an offset spatula, spread the filling over the cake to within 1-1/2 inches of the far short edge and to within 1/2 inch of the other edges. Distribute the pineapple over the cream cheese mixture. Be sure to coax the filling into the interior of the closer curled end so that when the cake is sliced there will be filling in the center of the spiral
Reroll the cake without the towel this time. The filling may squish out of the ends a bit, which is fine.