Preheat the oven to 350°F. Butter an 8-inch round or square baking pan.
In a medium bowl, toss together the strawberries with 1 tablespoon of the brown sugar, the lemon juice, and 1 teaspoon of the lemon zest. Allow to sit until the berries release some juice, 10 to 15 minutes.
In a second medium bowl, whisk together the flour, baking powder, and salt. In a large bowl, whisk together the buttermilk, vanilla, butter, and the remaining ⅔ cup brown sugar and 1 teaspoon lemon zest. Gradually mix in the flour mixture until smooth.
Spoon the batter into the prepared baking pan, spreading it in an even layer. Use a slotted spoon to spoon the strawberries over the top of the batter. Drizzle with about 2 teaspoons of the macerated strawberry juice. Bake until golden, set, and a toothpick inserted in the center comes out clean, 30 to 35 minutes. It should still be a bit soft and moist in the middle, with the corners of the cake starting to pull away from the edge of the pan.
Serve the spoon cake warm with ice cream or sweetened whipped cream.