Preheat the oven to 355 F (160 C)
Line a 10.5 x 15.5 inch sheet pan with parchment paper, and use a teaspoon of softened butter to butter the paper.
Sift the flour onto a piece of parchment paper and set aside.
Break the eggs into a heat-safe bowl of an electric mixer and add the sugar and vanilla.
In a pot that's big enough to fit the bowl, heat a few inches of water until hot.
Place the bowl in the hot water bath and mix gently until the egg reaches 100 degrees F (38 C) about the temperature of bathwater.
Attach the bowl to the mixer and beat at high speed using the whisk attachment for 5 minutes.
Turn down the speed to medium and slowly pour in the melted butter.
When the butter has been fully incorporated, turn down the speed to low and add the flour. Continue mixing until there are no lumps of flour, but be careful not to overmix. You may need to stop the mixer once and scrape down the sides.
Pour the cake batter into the prepared sheet pan and smooth off the top using an offset spatula or pastry knife. Smack the bottom of the pan against the countertop to force any large air bubbles to the surface.
Bake for 10-12 minutes, or until the cake is set in the center.
Let the cake cool on a rack until room temperature.
Slice the tops off of 4 strawberries and then cut them in half.
Using a whisk or electric mixer, whip 1 cup of heavy cream with 2 tablespoons of powdered sugar until it holds firm peaks. The cream should be stiff, but be careful not overwhip it, or it will get chunky.
Peel the parchment paper away from the cake.
Slice 2-inches (5cm) of cake off from the left side at an angle. This makes it easier to finish rolling the cake.
Spread the whipped cream all over the top of the sponge cake (you don't need to spread it on the cut surface).
Spread the strawberries in a line about 2-inches from the right edge.
Turn sheet, so the side with the strawberries is closest to you, and then use the parchment paper to roll the cake up and over the berries. Continue rolling the cake, being careful not the roll the parchment paper inside.
When you're finished rolling, wrap the extra parchment paper around the cake and then set the roll on a sheet pan with the seam at the bottom. Refrigerate for at least 1 hour.
When the cream has set, unwrap the cake onto a flat surface and then trim off both ends of the roll.
Dust the roll with powdered sugar and place the cake on your serving platter.
When you're ready to serve the cake, whip the remaining cream with the remaining powdered sugar until it forms soft peaks. You want the cream firm enough so it doesn't run all over the place, but it should still be loose enough to pour off of a spoon.
Slice the tops off of the remaining strawberries.
Drop dollops of whipped cream off of a spoon onto the top of the cake to make mounds of snow.
Top with the strawberries, and if you want, you can decorate your cake with sugar Santas and food-safe decorations like silver dragées.