Make the cornbread: Place a 9-inch cast-iron skillet in the oven and preheat to 400 degrees F. Whisk the cornmeal, flour, sugar, salt, baking powder and baking soda in a large bowl. Whisk in the milk, buttermilk, eggs and almost all of the melted butter, reserving about 1 tablespoon for brushing the skillet.
Carefully remove the hot skillet from the oven and brush the bottom and sides with the reserved melted butter. Pour the batter into the skillet, return to the oven and bake until a toothpick inserted into the center of the cornbread comes out clean, 20 to 25 minutes. Let cool 10 to 15 minutes, then sprinkle lightly with sugar for texture.
Meanwhile, make the jam: Cut the strawberries width-wise into 1/4-inch-thick slices (the slices should be thin enough to bend slightly). Using a vegetable peeler, remove the zest from 1/3 of the lemon with a light touch, leaving the pith behind; transfer to a large bowl. Squeeze the lemon juice into the bowl. Add the orange juice, salt, sugar and strawberries and toss. Cover and refrigerate until the cornbread is out of the oven.
Melt the butter in a large skillet over medium-high heat. Add the strawberry mixture and cook, stirring, until the juices reduce and thicken slightly, about 10 minutes. Slice the cornbread and serve with the strawberry jam.