Combine the cream cheese and 1/2 cup of the sugar in the bowl of a stand mixer fitted with a whisk attachment (or a large bowl if using a hand mixer) and beat on medium-high until smooth, about 3 minutes; scrape down the sides of the bowl as needed. Add 3 cups of the cream and 1 tablespoon of the vanilla and continue to beat, starting on medium speed and increasing to medium-high, until stiff peaks form, about 2 minutes.
Completely line a 9-inch springform pan with plastic wrap, leaving about a 2-inch overhang all around. Cut the doughnuts in half horizontally, making 2 thin circular halves. Line the side of the pan with 8 doughnut halves, standing them up vertically with the cut sides facing in.
Line the bottom with 4 halves, cut-side up; break 1 doughnut half into pieces to fill any gaps. Spread 1/3 of the whipped cream filling on top of the doughnuts. Repeat twice with the remaining doughnuts and filling (you will have 1 extra doughnut half to break up and fill gaps with). Loosely wrap the cake with the plastic wrap overhang and refrigerate overnight.
When ready to serve, unmold the springform pan and remove the plastic wrap. Combine the remaining 1/2 cup heavy cream, 1 tablespoon sugar and 1/4 teaspoon vanilla in the bowl of a stand mixer fitted with a whisk attachment (or a large bowl if using a hand mixer) and whip, starting on medium speed and increasing to medium-high, until stiff peaks form, about 1 minute. Transfer the mixture to a pastry bag and pipe around the edges of the cake as desired. Transfer the cake to a serving platter and decorate with sprinkles and birthday candles.