Preheat the oven to 350F and spray a Bundt pan very well with nonstick spray.
In a large bowl with an electric mixer on medium speed, beat the butter, oil and sugar until light and fluffy. Beat in the eggs one by one, along with the vanilla.
Wash and quarter the mandarin oranges and pulse in a food processor until mostly puréed — it will still be chunky, and that’s a good thing. In a bowl, stir together the flour, baking powder, baking soda and salt.
Add half the dry ingredients to the butter mixture, beating on low or stirring just until combined, then add about 1 1/2 cups of the orange pulp (don’t worry if it’s a bit more or less), then the remaining dry ingredients. Scrape into the bundt pan and bake for 45-55 minutes, until deep golden and springy to the touch. (If you like, stick a bamboo skewer in — it should come out without crumbs sticking to it.)
Let cool for about 20 minutes, then turn out onto a plate or wire rack while still warm. Whisk together 1 cup of the icing sugar, melted butter and mandarin juice (or milk/cream) until you have a drizzling consistency — add a bit more sugar if it’s too thin. Drizzle over the warm or cooled cake. Serves 16.