Make the Roasted Strawberry Filling: Preheat the oven to 350 degrees. Combine strawberries, sugar and salt in an 8 inch square baking dish. Roast in the oven for 20 minutes. Remove from the oven and stir briefly. Return to the oven and roast for an additional 20 minutes. Remove from the oven and allow it to cool slightly. Mash mixture with the back of a fork or a potato masher until it is a jam-like texture. Cover and refrigerate until ready to use. Filling should be completely cool before assembling the cake. (Filling can be made up to a week in advance).
Make the Vanilla Cream Filling: Whisk egg yolks, butter, cornstarch, half and half, sugar and salt together in a medium saucepan. Place pan over medium low heat and whisk constantly until mixture is smooth and thick, about 3-5 minutes. Strain mixture through a fine mesh strainer into a clean bowl. Stir in vanilla. Place a piece of parchment paper directly on top of the pastry cream, followed by a layer of plastic wrap (this prevents the custard from forming a skin). Refrigerate until you are ready to use. (Vanilla Cream Filling can be made 1-2 days in advance.)
Make the Cake: Preheat ovent to 350 degrees. Spray a rimmed baking sheet (12x18) with baking spray and line the bottom with parchment paper. Spray parchment with baking spray and use your fingers or a pastry brush to make sure the spray is distributed evenly around the top of the parchment paper and the pan.
Place the eggs, sugar, baking powder and salt in the work bowl of a stand mixer. Using the whisk attachment, mix on low speed until foamy, about 2-3 minutes. Scrape the bottom and sides of the bowl to make sure everything is incorporated. Increase the speed to medium and whip for 4 minutes. Increase the speed to medium high and whip until the mixture is thick and pale yellow in color, an additional 4-5 minutes. Decrease speed to medium low and add melted butter and vanilla.
Remove the work bowl and, using a fine mesh strainer, sift flour over the top of the mixture. Return the bowl to the mixer and mix on low speed until flour is just incorporated. Remove the work bowl and the whisk attachment. Use the attachment to whisk the mixture a couple of times by hand to make sure no pockets of flour remain.
Pour mixture into the prepared pan and even out the top. It will be a thin layer of batter. Transfer to the oven and bake until the cake is firm and springs back when touched, about 12 minutes.
Let the cake cool for 5 or 10 minutes in the pan. Run a knife along the edge of the pan and invert the cake onto a clean cotton dish towel (NOT terry cloth) that has been generously dusted with powdered sugar. Peel off the parchment paper that was on the bottom of the cake while it was baking. Starting with the short side of the rectangle, roll the cake in the dishcloth and allow it to rest for 15 minutes.
Carefully unroll the cake, and spread with the Vanilla Cream Filling, leaving a 1 inch border around the edges. It will be a thin layer. Gently top with 1 cup of the Roasted Strawberry Filling. Reroll the cake around the filling leaving the towel on the counter). Trim the ends and transfer to a serving plate (two spatulas, one on either side of the cake, works well for this). Allow to cool completely before dusting with powdered sugar and serving. The cake can be assembled and refrigerated for several hours prior to serving.