To prepare the cobbler, place the strawberries and rhubarb in a large, heavy saucepan. Add 1 tablespoon sugar. Stir over high heat until the sugar has started to dissolve, 1 to 2 minutes. Reduce the heat to medium and cook, stirring often, until the fruit has exuded some of its liquid, about 5 minutes.

In a large bowl, combine the fruit and all liquid with the remaining 4 tablespoons sugar, the flour, lemon zest, and salt. Cool slightly, pour into a lightly buttered 1-quart baking dish, and dot with the slivers of butter.

To make the topping, preheat the toaster oven to 400 degrees F. Combine the flour, sugar, baking powder, baking soda, and salt in a large bowl. Use a pastry blender or your fingertips to mix in the ginger and to cut in the butter until the mixture resembles the texture of small peas. Quickly stir in the buttermilk to form a soft dough. Turn out onto a well-floured work surface, knead into a ball, and roll out to a thickness of about 1/2 inch. Use a 1 1/2-inch cookie cutter to cut out 4 biscuits. Gather the trimmings and roll out again. Cut out 2 more biscuits.

Arrange the biscuits snugly over the fruit. Brush lightly with additional buttermilk, sprinkle with sugar, and bake until the cobbler is bubbly and the biscuits are golden brown, 25 to 30 minutes. Serve hot.