In a food processor, puree 1 pound strawberries with 1/2 cup granulated sugar until smooth. Transfer to a medium saucepan and bring to a boil, then reduce heat to medium and cook, stirring frequently, until darkened, thick, and syrupy, about 10 minutes. Strain through a fine-mesh sieve, pressing on solids to remove as much liquid as possible (you should have 1 cup). Refrigerate, covered, until room temperature, about 1 hour (or transfer to a bowl set in an ice-water bath and stir occasionally until cool to the touch, about 10 minutes).
Combine egg whites, remaining 1 1/4 cups granulated sugar, and salt in a heatproof standmixer bowl set over (not in) a pot of simmering water, whisking constantly until sugar is dissolved and mixture is warm. (It should feel completely smooth when rubbed between your fingertips.) Transfer bowl to mixer fitted with the whisk attachment. Starting on low speed and gradually increasing to medium-high, beat until stiff, glossy peaks form. Continue beating until bottom of bowl is cool to the touch, about 10 minutes.
Reduce speed to medium-low and beat in butter, a few tablespoons at a time, until combined, scraping down bowl as needed. Slowly add strawberry puree and vanilla, beating until combined. (If buttercream appears curdled, increase speed to medium-high and beat until smooth.)
Slice 1 pound strawberries lengthwise, 1/4 inch thick (you should have about 2 1/2 cups). Cut cake horizontally into 3 even layers with a serrated knife. Transfer bottom layer, cut-side up, to a cake plate or stand lined with parchment strips. Spread top evenly with 3/4 cup buttercream and half of sliced strawberries, overlapping berries to fit. Spread another 3/4 cup buttercream onto one side of center cake layer and place over first cake layer, frosted-side down. Press gently to adhere, being careful not to allow strawberries to slide out. Spread top with another 3/4 cup buttercream; top evenly with remaining sliced strawberries. Spread another 3/4 cup buttercream onto cut side of top cake layer; place on center cake, frosted-side down, pressing gently to adhere.
Using a small offset metal spatula, spread remaining buttercream evenly over top and sides of cake. To create decorative sides, hold spatula vertically against cake and press one long side gently into frosting (do not go deeper than 1/4 inch, so you don't get down to cake), then pull spatula directly upward in one motion. Repeat around entire cake, then remove parchment strips. Top cake with remaining 1 pound strawberries, cutting some in half, if desired. Dust lightly with confectioners' sugar; serve. This cake is best assembled, kept at room temperature, and served the same day, but it can be refrigerated, uncovered, up to 1 day; bring to room temperature before serving.