Melt 4 tablespoons unsalted butter in a small saucepan over medium heat, swirling occasionally, until golden and nutty-smelling, about 4–5 minutes. Pour into a large mixing bowl and set aside to cool completely while you handle everything else.
Preheat the oven to 400°F. Line a small baking sheet with parchment. Toss 1 lb fresh strawberries, hulled and halved if large, with 1 tablespoon granulated sugar, 1 tablespoon coarse sugar and 1 teaspoon balsamic vinegar until evenly coated. Arrange cut-side up. Roast for 10 minutes until glossy and slightly concentrated. Let cool slightly, then gently drain off any excess juice. Reduce oven temperature to 350°F.
Grease a 9-inch round cake pan with butter, then line the bottom with a round of parchment paper.
In a small saucepan over medium heat, melt 4 tablespoons unsalted butter (for caramel). Stir in ⅔ cup packed light brown sugar and ¾ teaspoon ground cardamom (for caramel). Cook for 1–2 minutes, stirring frequently, until smooth and glossy. Remove from heat. Pour into the prepared pan and spread evenly.
Spread the roasted strawberries over the caramel in a single even layer. Pack them in; Set the pan aside.
In a large bowl, whisk 2 tablespoons unsalted butter into the cooled brown butter until smooth. In a separate medium bowl, whisk together 1 cup all-purpose flour, ½ cup light rye flour, 1¾ teaspoon baking powder, ½ teaspoon baking soda, ¾ teaspoon fine salt, and ½ teaspoon ground cardamom. Set aside.
Add ⅔ cup coconut sugar to the egg yolks and whisk for 1–2 minutes, until slightly lighter and well combined. Coconut sugar dissolves more slowly than white sugar, so this extra mixing time helps create a smoother batter and a softer crumb.
Whisk in 1 cup full-fat sour cream, 3 tablespoons milk, 1.5 teaspoons vanilla extract, and 1 teaspoon lemon zest until smooth.
Pour into the brown butter mixture and whisk until fully combined.
Add the dry ingredients to the wet mixture and stir gently until just combined. Do not overmix.
Beat the egg whites with a hand mixer or whisk until soft peaks form, airy and holding their shape, not stiff. Fold into the cake batter in 2 additions. Work gently to keep the air in.
Carefully spoon the batter over the strawberry layer, then smooth the surface with an offset spatula. Tap the pan once on the counter to release air pockets.
Bake at 350°F for 38-41 minutes until the top is deep golden and a tester inserted in the center comes out with a few moist crumbs. If the top browns too quickly, tent loosely with foil during the last 10 minutes.
Let the cake rest in the pan for 10–15 minutes, no longer, or the caramel will seize. Gently run a knife around the edges, place a serving plate on top, and flip in one confident motion. Leave the pan on for 1 minute before lifting. Peel away the parchment. Serve warm or at room temperature with powdered sugar, lightly sweetened śmietana, whipped cream, or crème fraîche.
Notes
Roast the strawberries before assembling the cake. This concentrates their flavor and helps prevent excess moisture from making the cake soggy.Whisk the coconut sugar and egg yolks together for a full minute or two. This helps dissolve the sugar and creates a smoother, more tender crumb.