Grease a 9-inch round cake pan with butter and dust with flour, tapping out the excess. Set aside. In a medium bowl, whisk together the flour, baking powder and salt.
Beat the butter in a large bowl until fluffy, about 2 minutes. Add the sugar and continue to beat until very fluffy, about 2 minutes longer. In a small bowl, whisk the eggs and cream. Add half the flour mixture and half the egg mixture to the creamed butter and beat until combined. Repeat with the remaining flour and egg mixture. Add the lemon zest and vanilla and beat the batter on medium-high speed for an additional 1 to 2 minutes.
Transfer the batter to the prepared cake pan, spread the top evenly and bake until a toothpick inserted into the center comes out clean and the top sprigs back to the touch, 40 to 45 minutes. Cool in the pan 10 minutes, and then turn out onto a wire rack and cool completely.
While the cake bakes and cools, make the filling: Combine the rhubarb with the sugar and 1/4 cup water in a medium pot. Bring to a simmer over medium heat and continue to cook, stirring, until the rhubarb cooks down into a thick puree, 18 to 20 minutes. Cool completely, and then fold in half the strawberries. Stir the sugar into the remaining berries.
For the whipped cream: When ready to assemble the cake, whip the cream with the sugar and vanilla until stiff peaks form.
Using a serrated knife, slice the cake in half horizontally and place the bottom half on a cake stand or serving plate, cut-side up. Spread with 1 cup of the whipped cream, stopping about 1/2 inch from the edge of the cake and top with the strawberry-rhubarb mixture. Place the second half of the cake, cut-side down, over the strawberry-rhubarb filling and spread the remaining whipped cream over top, going close to the edge of the cake. Garnish with the remaining sliced strawberries.
NOTE: To make this cake ahead, you can fully assemble and chill up to 4 hours ahead of time without the sliced strawberry and sugar garnish. Top with the sliced strawberries before serving.