Preheat the oven to 350°F. Line a 9 X 5-inch loaf pan and set aside.
In a medium bowl, whisk together the all-purpose flour, poppy seeds, baking powder, baking soda, and salt. Set aside.
In a large mixing bow, add the sugar and lemon zest. Rub the ingredients together between your fingertips until fragrant. Add the softened butter and mix with a hand mixer until light and fluffy, 2 to 3 minutes. Mix in the oil, vanilla, and eggs - one at time. Mix until blended.
Add in half of the dry ingredients and mix on low until just combined. Add the yogurt and lemon just then mix until smooth. Add the remaining dry ingredients and mix on low speed until just combined. Use a rubber spatula to fold the batter together until smooth.
Scoop the batter into the prepared pan and smooth it out with the rubber spatula. Bake for 45 to 50 minutes, or until a wooden skewer inserted into the center of the cake comes out clean. Cool the cake on a wire rack.
Make the Drizzle
Stir together the powdered sugar and lemon juice until smooth. It should thicker than simple syrup but much thinner than frosting.
Poke holes into the top of the cake and pour the drizzle all over the top. The drizzle with dry and leave a crackly finish.
Notes
Store the lemon poppy seed cake at room temperature for up to 2 days. Loosely cover with plastic wrap or tent with foil. Store leftovers in the refrigerator. The cake is best served within 3 days.