Make the lemon cake
Preheat oven to 350°F. Combine granulated sugar and lemon zest in a large bowl, and rub with fingers to disperse lemon oil until sugar feels moist, about 1 minute. Whisk in olive oil. Add eggs; whisk until well combined. Add milk, vanilla, and salt; whisk until sugar is dissolved, about 2 minutes. Whisk in flour and baking powder just until combined, and then whisk 20 turns around the bowl. Lightly coat a 13- x 9-inch metal baking pan with cooking spray; pour batter into pan. Bake in preheated oven until a wooden skewer inserted in center comes out clean, 30 to 35 minutes. Let cake cool in pan on a wire rack 10 minutes. Invert cake onto wire rack, and poke cake all over with a wooden skewer or fork 15 to 20 times. Stir together honey and lemon juice in a small bowl until smooth. Brush or drizzle syrup over warm cake. Let cake cool completely, about 1 hour.
Make the cream cheese frosting
Combine cream cheese, lemon zest and juice, and salt in a large bowl. Beat with an electric mixer on medium-high speed until smooth, about 30 seconds. Add butter, and beat on high speed until pale and fluffy, about 1 minute, stopping to scrape sides of bowl as needed. Reduce speed to low, and gradually add powdered sugar, beating until incorporated. Increase speed to high, and beat until fluffy, about 2 minutes.
Transfer cake to a parchment paper–lined baking sheet. Spread 1 1/2 cups frosting evenly over top of cake. Refrigerate, uncovered, until frosting hardens, about 15 minutes. Cut cake in half crosswise, and stack halves. Spread remaining frosting on top and sides of cake. Refrigerate cake, uncovered, 30 minutes, or cover and refrigerate up to 3 days. Serve cake chilled.
Assembled cake can be covered and stored in refrigerator up to 3 days.