Peel and chop 2 of the beets, then puree them in a mini food processor. Peel and shred the remaining beet.
Preheat the oven to 350°. Butter a 10-by-15-inch jelly roll pan and line it with parchment paper. In a small bowl, whisk the flour, baking powder, baking soda and salt. In a large mixing bowl, beat the eggs with the sugar at high speed until tripled in volume, about 5 minutes. Beat in the oil and zest. Fold in the pureed and shredded beets, then gradually fold in the dry ingredients. Spread the batter in the prepared pan, smoothing the top. Bake the cake for about 20 minutes, until a toothpick inserted in the center comes out clean.
Bring 1 inch of water to a boil in a saucepan. In a stainless steel bowl, whisk the yolks and sugar until thickened. Whisk in the warm milk and cream. Set the bowl over (not in) the simmering water and whisk over moderate heat until the custard is very thick, about 8 minutes. Chill the custard in an ice bath, then stir in the bourbon.
Cut the cake into squares and layer with the custard. Top with whipped cream; serve.