Prep the pans. Preheat the oven to 350 degrees F. Grease two 8 inch round cake pans and line the bottoms with parchment paper. Whisk 2 cups flour, 2 1/2 tsp baking powder, and 1/4 tsp salt in a bowl and set aside.
Cook the frosting base. Whisk 6 1/2 tbsp flour and 1 1/3 cups milk in a saucepan until smooth. Cook over medium heat, whisking constantly, for 3 to 5 minutes until thick like pudding. Press plastic wrap against the surface and cool to room temperature, about 45 minutes.
Warm the butter and milk. In a small saucepan, heat 1 cup milk and the cubed butter over medium-low heat for about 3 minutes until the butter melts and the milk steams. Do not boil. Cool 5 minutes, then stir in the vanilla, oil, and lemon extract.
Whip the eggs. Beat the eggs and 1 1/2 cups sugar on high speed for 4 minutes until pale and thick. Beat in 3 tbsp lemon zest and 1/3 cup lemon juice.
Combine. Pour in the warm milk mixture on low speed, then fold in the flour mixture in two additions until no dry streaks remain. The batter will be thin.
Bake. Divide the batter between the pans and bake 25 to 30 minutes until the tops are golden and a toothpick comes out clean.
Cool. Cool in the pans 10 minutes, turn out onto wire racks, and cool completely, about 1 hour.
Whip the frosting. Beat the softened butter on medium-high for 3 minutes until pale. Add the cooled flour mixture a spoonful at a time, then the zest and juice, and beat 2 more minutes until fluffy.
Assemble. Spread frosting over one layer, top with the second layer, and frost the top and sides.