Let’s break down this detailed process:

Make the Cake:

1. Prepare Pans and Preheat Oven: Grease and flour two 9-inch round cake pans (with high sides) or three 8-inch round cake pans (with high sides). Line the bottoms with parchment paper rounds and grease/flour the parchment too. Preheat your oven to 325°F (165°C).
2. Cream Butter and Sugar: In the bowl of your stand mixer (or large bowl with hand mixer), whip the softened butter and granulated sugar together on medium-high speed until very fluffy and creamy. (The recipe implies this takes time, aiming for dissolved sugar – likely 5-7 minutes).
3. Add Eggs and Yolks: Add the eggs and egg yolks one at a time, mixing well after each addition. Scrape down the sides of the bowl regularly.
4. Add Oil and Vanilla: Mix in the oil and vanilla bean paste on low speed until just combined.
5. Prepare Dry and Wet: Measure the buttermilk into a liquid measuring cup and set aside. Sift the flour, baking powder, baking soda, and salt into a separate bowl and combine with a wire whisk.
6. Alternate Additions: With the mixer running on low speed, add the dry ingredients and the buttermilk alternately to the butter mixture. Start with 1/3 of the dry ingredients, then half the buttermilk, another 1/3 dry, the remaining buttermilk, and finally the last 1/3 dry ingredients. Mix only until just combined after each addition.
7. Final Mix: Turn the speed to medium-low and mix until the batter is smooth, about one minute. Do not overmix.
8. Bake: Pour the batter evenly into the prepared cake pans. Bake at 325°F (165°C) for about 29-34 minutes, or until a toothpick inserted into the middle comes out clean.
9. Cool: Cool the cakes in the pans on cooling racks for 5-7 minutes, then turn them out onto the racks to cool completely.
10. Chill Layers: Once cool, wrap the cake layers individually in plastic wrap and refrigerate for at least 1 hour (or longer) before frosting. Chilled cakes are much easier to frost.

Make the Frosting:

1. Beat Butter: Beat the softened butter in the mixer bowl on high speed until light, fluffy, and smooth.
2. Add Powdered Sugar: Sift the powdered sugar. With the mixer on low, add it slowly, one cup at a time, mixing after each addition until incorporated.
3. Add Flavor and Cream: Add the vanilla bean paste and salt. Slowly add the heavy cream one tablespoon at a time, mixing until incorporated and the frosting reaches a spreadable consistency.
4. Final Whip: Mix the frosting on medium-high speed for 1-2 minutes until very smooth and spreadable.

Assemble and Decorate:

1. Prepare Frosting: Reserve about 1 1/2 cups of frosting for the crumb coat. Place the rest in a piping bag fitted with a star tip (if making rosettes). Set aside.
2. Level Cakes: Trim the cooled cake layers (if necessary) to level them using a serrated knife.
3. First Layer: Place one cake layer, right side up, on a plate or cake board. Add a large dollop of frosting (from the main batch, not the crumb coat reserve) on top and spread it out evenly to the edges.
4. Second Layer: Place the next cake layer on top (upside down for a flat top surface). If making a three-layer cake, repeat the frosting step. Place the final layer upside down.
5. Crumb Coat: Add the reserved 1 1/2 cups of frosting to the top of the cake. Spread it thinly over the top and down the sides, smoothing it out to fill any gaps and create a smooth, thin layer. It’s okay if cake shows through.
6. Chill Crumb Coat: Place the cake in the fridge to chill for about 45 minutes until the crumb coat is firm.
7. Final Frosting (Rosettes): Remove the cake from the fridge. Using the piping bag with the star tip:
   • Start piping rosettes on the side of the cake, beginning at the very bottom. Pipe a full row.
   • Offset the second row so the rosettes fill the gaps above the first row. Continue up the sides.
   • On top, pipe rosettes around the outside edge first, then work your way inward, offsetting rows to fill gaps. Pipe rosettes close together, ending the swirl of each rosette in a gap where the next one will start to hide the tail.
   • (Alternative): If not piping rosettes, simply apply the remaining frosting smoothly over the chilled crumb coat using an offset spatula.
8. Final Chill: Chill the finished cake for another 30 minutes to an hour to set the frosting.
9. Serve: Let the cake sit at room temperature for about 15 minutes before slicing and serving (this allows the buttercream to soften slightly).

This detailed process ensures a beautiful and delicious cake!