Preheat oven to 350°F (177°C). Grease and lightly flour three 9-inch cake pan.
On medium-high speed, add 1 egg at a time, beating well after each addition. Beat in the vanilla extract. Scrape down the sides and up the bottom of the bowl with a rubber spatula as needed.
There is about 8 cups of batter total, so each layer will be 2 and 2/3 cups. Pour/spread batter evenly into prepared cake pans.
In a large bowl using a hand-held mixer or stand mixer fitted with a whisk or paddle attachment, beat the butter on medium speed until creamy - about 2 minutes.
With the mixer running on low, add 5 and 1/2 cups confectioners' sugar, 5 tablespoons heavy cream, and the vanilla extract. Increase to high speed and beat for 3 full minutes.
Add 1/2 cup more confectioners' sugar if frosting is too thin, 1 more tablespoon cream if frosting is too thick.
First, using a large serrated knife, slice a thin layer off the tops of the cakes to create a flat surface. Discard or save and eat later.
Place 1 cake layer on your cake stand, cake turntable, or serving plate. Spread 1 and 1/2 cups of frosting evenly on top. Spread the raspberry filling right on top of the frosting (several spoonfuls) until the frosting is covered with the filling. Use your own judgement on how much to use.
Top with 2nd cake layer and evenly cover the top with another 1 and 1/2 cups of frosting and more raspberry filling.
Use a bench scraper to smooth out the frosting on the sides of the cake.
Refrigerate the cake for 2-3 hours or up to 1 day before slicing and serving. If chilling for more than a few hours, loosely cover with plastic wrap after 1 hour of refrigeration (the frosting will "set" after about 1 hour and the plastic wrap won't really ruin your frosting at that point). And if you're chilling for more than a few hours, I suggest decorating with garnishes right before serving.
6 hours of total time includes resting time for the cake and raspberry filling to completely cool.