If making the strawberry reduction: place all the ingredients in a pot and cook over low medium heat until the strawberries break apart and you have a thick mixture, about 15-20 minutes. Blend to puree. If you don’t want seeds, pour the puree through a fine mesh sieve. Let cool completely before adding to the cake.
Preheat the oven to 350 F. Grease and flour a 9 inch square or circle baking pan. Put the eggs in a bowl of hot tap water to bring them to room temperature.
Set the cold butter in a heatproof bowl and microwave for 1 minute, until mostly melted.
Pour in the vanilla and salt and whisk to combine. Pour in the sugar and whisk very well, for a full 60 seconds.
Add the eggs, one by one, whisking for a good 30 seconds before adding the next. Once all the eggs are in, whisk vigorously for a full 45 seconds.
If the flour is lumpy, sift it into the batter (if it’s not - just add it in). Add the baking powder and stir to combine in the dry ingredients.
Gently whisk in the yogurt.
Spread half of the batter into the bottom of the prepared pan. Either add dollops of strawberry reduction or arrange enough thinly sliced strawberries to entirely cover the cake batter, laying them flat (see photos for reference).
Spread the remaining half of the cake batter over the strawberries in an even layer. Thinly slice another layer of strawberries to entirely cover the top of the cake. Spread 2-3 tablespoons of coarse sugar over the strawberries.
Bake the cake until a toothpick inserted into the center of the cake comes out clean (it might show some jammy strawberries but it shouldn’t show any gooey cake batter), about 45-55 minutes. Most of the strawberries you placed on top will have sunk and now be layered into the cake but some will still show in the center.
Let the cake cool at room temperature for 20 minutes. You can try to flip it onto a plate but it will be less messy if served straight from the pan.
Serve with whipped cream or ice cream. Store leftovers in an airtight container either at room temperature for 1-2 days or in the fridge for 2-3 days.