Preheat oven to 350F. Grease two 8-inch cake pans and line bottoms with parchment paper.
In a large bowl, sift flour, baking powder, baking soda and salt. Mix well.
In the bowl of an electric mixer fitted with the paddle attachment, mix eggs and sugar until well combined. Gradually add oil and vanilla, mix until just blended.
With the mixer on low, slowly add half the flour mixture; mix until just blended. Add the buttermilk, mix until just blended. Slowly add remaining flour mixture, scraping down sides of the bowl with a spatula, as necessary, until just blended.
Divide the batter and pour into the prepared pans, smoothing the surface with a spatula. Bake 20 to 25 minutes or until a toothpick inserted into the cake comes out clean.
Place the cakes (in the pans) on a wire rack to cool. Then invert the cakes onto a greased rack. Let cool completely.
Reserve one of the cake layers for future use. Place another cake layer in the bottom of an 8-inch springform pan.
To prepare strawberry puree for the ice cream, place strawberries and 3 tablespoons sugar in a saucepan and bring to a boil. Then reduce heat and simmer until sauce starts to thicken, about 10 minutes. Let cool slightly, then puree the mixture in a blender until smooth. Refrigerate until ready to use.
To prepare the ice cream, in a medium saucepan, combine milk, heavy cream, 1/4 cup sugar and salt. Bring mixture to simmer, stir until sugar is dissolved. Remove from heat and set aside.
Whisk egg yolks and remaining sugar in a medium bowl until pale, about 2 minutes. Gradually whisk in half of the warm cream mixture, mix until blended. Stir yolk mixture into the remaining cream mixture. Cook over medium heat, stirring constantly, until thick enough to coat a wooden spoon, about 2 to 3 minutes.
Add cream cheese and stir until completely melted and mixture is smooth.
Strain custard into another medium bowl. Let cool over a bowl of ice water, or chill in the refrigerator, stirring occasionally.
Whisk in the strawberry puree.
Process the mixture in an ice cream maker according to manufacturer’s instructions. Add graham cracker crumbs in the last minute of processing. Spread the soften ice cream over the cooled cake base. Place the pan in freezer to firm up, at least 30 minutes.
To prepare topping, whip the heavy cream and sugar until stiff peaks form. Spread the whipped cream over the ice cream layer. Smooth the top and garnish with sprinkles.
Store in freezer for at least 4 hours to firm up the ice cream. Before serving, place a hot towel around the outside of the springform pan for 30 seconds to release the ring of the pan.