Preheat oven to 350°.

Brush an 8-in. square pan with 1 Tablespoon of oil and dust with 2 Tablespoons of flour.

Take one or two of the tangerines and using a zester or small grater, grate the zest until you have a tablespoon of it, taking care not to get the bitter white pith into it. Reserve.

Cut the zested tangerines in half and juice them through a fine strainer until you have 1/2 cup of juice. You may need a few more tangerines. Reserve.

Take five of the tangerines and thinly slice them crossways into rounds, discarding the seeds and ends. Reserve the slices and any juice from them.

In a medium bowl, combine the flour, baking powder, salt, baking soda, ground cardamom, and half (3 Tablespoons) of the sesame seeds.

Place the 1/2 cup oil and the 1/2 cup granulated sugar in a large bowl and with an electric mixer on medium- high speed, beat until well blended, about one to two minutes. Add the eggs and beat another one to two minutes.

Move the mixer speed to low and add in the tangerine zest and juice. Then add the flour mixture and beat on medium-low speed until the batter is smooth, about 1 minute more. Scrape the batter into the prepared 8" pan. Set aside 1 teaspoon of sesame seeds. Sprinkle the remaining sesame seeds over batter.

Bake the cake until the top is golden and a toothpick inserted in center comes out clean, about 30 to 35 minutes. Let the cake cool on a rack 10 for minutes then loosen the cake from sides of pan and invert onto a cooling rack. Invert the cake, sesame-side up, onto another rack and let cool completely.

In a medium saucepan, bring the 3/4 cup granulated sugar and 3/4 cup water to a boil over medium-high heat. Add the sliced tangerines and any juice and bring to a low simmer. Reduce the heat continue to simmer, stirring occasionally, until the tangerine skin is translucent and tender, about 30 minutes. Let cool completely.

Select the most beautiful of the candied tangerine slices to garnish the top of the cake and set those aside.

Take some of the less beautiful pieces and chop them into about 2 Tablespoons. Put the chopped candied tangerines in a bowl with the whipping cream. Beat with a mixer on medium speed until cream holds a soft peak. If the cream is not sweet enough you can add a teaspoon or two of the syrup.

Split the cake horizontally with a serrated knife to make two layers. Set the bottom half on a platter and spoon the whipped tangerine cream on top. Scatter three-quarters of sliced candied tangerines with a little syrup over the whipped cream layer.

Set the top of the cake on top of the whipped cream (sesame seed side up) and sift some powdered sugar on top.

Arrange the beautiful reserved slices of candied tangerine on top of the cake and sprinkle with the remaining sesame seeds and more powdered sugar.