Preheat the oven to 350-degrees. Spray a 13x9x2-inch (preferably) glass pan. Line the bottom with parchment, and spray the parchment. Set aside.
In the bowl of a stand mixer, fitted with the paddle attachment, beat the cream cheese and sugar on medium-low speed until light and fluffy, stopping and scraping the bowl as needed. Add the oil and vanilla and continue to mix until fully incorporated, scraping the bowl as necessary. On low speed, add the eggs and yolks, one at a time, stopping and scraping the bowl after each addition.
Continuing on low speed, add the flour in three installments (I like to add a bit more than a third of the flour in the first installment), alternating with the club soda, and beginning and ending with the flour. Scrape the bowl down as needed, but definitely after the second flour installment. Remove the bowl from the mixer when there is still visible flour in the batter, and use a rubber spatula to finish mixing the batter.