Preheat the oven to 350°F. Pour the melted butter into a 7 1/2- by 12-inch (or similar sized) rectangular baking dish. Arrange the berries, cut-side up, in a single layer on top of the butter. Sprinkle with the 1/3 cup of sugar and set aside.
Put the (room-temperature) butter, sugar, vanilla, and salt in the workbowl of a food processor. Blitz until creamy -- about 10 seconds. Add the eggs and the lemon zest, and process until well incorporated -- about 10 seconds. Add 1/3 of the flour and half of the milk, pulsing the machine until the flour disappears into the batter. Then pulse in another 1/3 of the flour and remaining milk. Pulse in the remaining flour.
Pour the thick batter over the strawberries. Use a spatula or the back of a spoon to spread the batter as evenly as possible over the berries without dislodging them.
Bake in the preheated oven until the cake turns spotty brown and is just springy to the touch -- 30-35 minutes. Let cool on a wire rack for 5-10 minutes. Then run a knife around the edge of the cake. Unmold while still hot onto a baking sheet or serving platter. Serve warm or at room temperature. If you wish, top each serving with a dollop of vanilla ice cream, crème fraiche, or mascarpone.