Line a large strainer with two layers of cheesecloth and place ricotta on top. Cover with plastic wrap and strain in the refrigerator for 4–12 hours to remove excess moisture.
To make the cannoli filling: In a mixing bowl, combine strained ricotta, 3/4 cup mascarpone, and 1/3 cup powdered sugar. Gently mix by hand or on low speed until just combined—do not overmix. Refrigerate for 2–3 hours to thicken.
Preheat oven to 350°F (176°C). Grease and line three 8-inch round cake pans with parchment paper.
In a medium bowl, whisk together flour, baking powder, baking soda, cinnamon, and salt. Set aside.
In a large bowl, cream together butter, oil, granulated sugar, brown sugar, and 1 tbsp vanilla extract on high speed for 3 minutes until light and fluffy.
Add eggs one at a time, mixing just until incorporated after each. Scrape down the bowl as needed.
Add half of the dry ingredients and mix until mostly combined.
Slowly mix in milk and sour cream until fully incorporated. The batter may look slightly curdled—this is normal.
Add remaining dry ingredients and mix just until smooth. Do not overmix.
Divide batter evenly among prepared pans. Bake for 20–24 minutes or until a toothpick inserted in center comes out clean with a few moist crumbs.
Let cakes cool in pans for 3 minutes, then turn onto wire racks to cool completely.
For the buttercream: Beat room temperature butter and cool mascarpone cheese on low for 1 minute until smooth. Gradually add 8 cups powdered sugar in two batches, mixing just until combined. Add 1 tsp vanilla and mix gently. Add remaining 1 cup powdered sugar if needed for consistency. Do not overmix.
Level the cooled cakes with a serrated knife. Place first layer on a cake board. Pipe a ring of buttercream around the edge to create a dam, then spread half of the chilled filling into the center.
Add second layer, pipe another dam, and add remaining filling. Top with third cake layer.
Apply a thin crumb coat of buttercream and refrigerate for 1–2 hours or freeze for 30 minutes until firm.
Frost the outside with remaining buttercream. Press 1 cup of mini chocolate chips onto the bottom edge of the cake.
Pipe remaining buttercream around the top edge using a large star tip (e.g., Ateco 847) for a decorative finish.
Refrigerate until ready to serve. Store leftovers in an airtight container in the fridge for up to 4 days.