Preparing the Cake Layers:

1. Preheat the Oven:
   Set your oven to 350°F (177°C).

2. Prepare the Pans:
   Grease two 8-inch round cake pans and line the bottoms with parchment paper.

3. Combine Dry Ingredients:
   In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt.

4. Cream Butter and Sugar:
   In a large mixing bowl or stand mixer, beat the room-temperature butter and granulated sugar until light and fluffy, about 3-4 minutes.

5. Add Oil and Eggs:
   Mix in the vegetable oil until combined. Add the eggs one at a time, beating well after each addition.

6. Incorporate Dry Ingredients and Milk:
   Alternate adding the dry ingredient mixture and the milk to the butter-sugar mixture, beginning and ending with the dry ingredients. Mix until just combined; avoid overmixing.

7. Bake the Cakes:
   Divide the batter evenly between the prepared pans. Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean.

8. Cool the Cakes:
   Allow the cakes to cool in the pans for 10 minutes, then remove them from the pans and place on a wire rack to cool completely.

Preparing the Lemon Syrup:

1. Heat the Ingredients:
   In a small saucepan, combine the granulated sugar and fresh lemon juice. Heat over low heat, stirring occasionally, until the sugar is fully dissolved.

2. Cool the Syrup:
   Remove from heat and let the syrup cool slightly before using.

3. Soak the Cake Layers:
   Using a pastry brush or a spoon, evenly drizzle the lemon syrup over both cake layers while they are still warm. Let the cakes absorb the syrup completely before assembling.

Preparing the Lemon Mascarpone Cream Filling:

1. Whip the Cream:
   In a chilled bowl, whip the cold heavy whipping cream to stiff peaks. Set aside.

2. Combine Mascarpone and Flavorings:
   In a separate bowl, mix the room-temperature mascarpone cheese, powdered sugar, lemon zest, and fresh lemon juice until smooth.

3. Fold in Whipped Cream:
   Gently fold the whipped cream into the mascarpone mixture until fully incorporated. Refrigerate until ready to use.

Preparing the Crumb Topping:

1. Preheat the Oven:
   Set your oven to 350°F (177°C).

2. Toast the Flour:
   Spread the all-purpose flour on a baking sheet and bake for 5 minutes to eliminate the raw flour taste. Let it cool slightly.

3. Combine Ingredients:
   In a bowl, whisk together the toasted flour and powdered sugar. Cut in the cold diced butter until the mixture resembles coarse crumbs. Stir in the lemon zest and vanilla extract.

Assembling the Cake:

1. Level the Cakes:
   If necessary, level the tops of the cooled cakes using a serrated knife.

2. Apply Filling:
   Place one cake layer on a serving plate. Spread a generous amount of the lemon mascarpone cream filling on top.

3. Add Second Layer:
   Place the second cake layer on top. Spread the remaining filling over the top and sides of the cake.

4. Apply Crumb Topping:
   Gently press the crumb topping onto the top and sides of the cake.

5. Chill Before Serving:
   Refrigerate the assembled cake for at least 1 hour before serving to allow the flavors to meld and the filling to set.