Make cake: Preheat oven to 350°. Brush an 8-in. square pan with oil and dust with flour. In a medium bowl, combine remaining 11/2 cups flour, the baking powder, salt, baking soda, cardamom, and 3 tbsp. sesame seeds.
In a large bowl with a mixer on medium- high speed, beat remaining 1/2 cup oil with the granulated sugar until well blended, 1 to 2 minutes. Add eggs and beat until slightly bubbly, 1 to 2 minutes. Incorporate tangerine zest and juice on low speed until blended. Add flour mixture and beat on medium-low speed until batter is smooth, scraping inside of bowl as needed, about 1 minute more. Scrape batter into prepared pan. Set aside 1 tsp. sesame seeds. Sprinkle remaining sesame seeds over batter.
Bake cake until golden and a toothpick inserted in center comes out clean, 30 to 35 minutes. Let cake cool on a rack 10 min-utes. Loosen cake from sides of pan and invert onto rack. Invert cake, sesame- side up, onto another rack and let cool completely.
Meanwhile, make topping: In a 11/2- to 2-qt. saucepan, bring granulated sugar and 3/4 cup water to a boil over medium-high heat. Add tangerines and any juice and bring to simmering. Reduce heat and simmer, stirring occasionally, until skin is translucent and tender, 30 to 45 min- utes. Let cool completely (if you like, set pan in ice water to speed it up).
Chop 2 tbsp. candied tangerines and put in a bowl with cream. Beat with a mixer on medium speed until cream holds a soft peak. Split cake horizontally with a serrated knife. Set bottom half on a plat- ter and spoon cream on top. Scatter three-quarters of candied tangerines with a little syrup over cream. Set cake top over cream and sift on some pow- dered sugar. Toss remaining candied fruit with reserved 1 tsp. sesame seeds and arrange over cake.
MAKE AHEAD The cake, up to 1 day, airtight at room temperature. The candied tangerines, up to 1 week, chilled airtight; bring to room temperature before using.
*Find olio nuovo such as California Olive Ranch Limited Reserve at well-stocked supermarkets.