Grease and flour a 6 to 8-cup bundt pan and set aside.
Blend the cream cheese with the 1/4 cup granulated sugar and the vanilla.
Scrape the bowl; add the egg and mix well. Set aside while you prepare the cake batter.
Sift the flour, the 1-1/3 cups granulated sugar, baking soda, and salt into a mixing bowl; whisk briefly to blend.
Whisk in the oil, lemon zest, and about half of the buttermilk until smooth. This first addition will prevent lumps from forming; do not be tempted to pour in all of the liquid at once.
Whisk in the remaining buttermilk, mixing only long enough to combine.
Pour half of the cake batter into the prepared bundt pan.
Dollop the cream cheese mixture over the batter, trying to center it in the pan so it doesn't reach either edge.
Cover with the remaining batter.
Bake for about 55 minutes, until the top is golden brown and slightly firm to the touch. If you check it with a toothpick or cake tester, there should be only traces of cream cheese and no loose batter.
Cool the cake in the pan for 15 minutes; invert onto a rack to cool completely.
To add the final touch, stir the lemon juice into the confectioners' sugar until smooth. Drizzle the glaze over the cake in a random pattern.