For the strawberry base: Put the strawberries in a bowl, add the sugar and gently stir to combine. Let the berries sit at room temperature for 30 minutes, until they start to soften and release their juices.
Transfer the mixture to a medium nonreactive saucepan, add the jam and cook over medium-high heat until slightly softened. Coarsely mash the berries using a potato masher or the back of a fork and continue cooking, stirring occasionally, until the mixture thickens to a jam-like consistency, about 20 minutes. Remove from the heat and stir in the lemon juice. Transfer to a bowl, cover and refrigerate until cold and thickened further, at least 2 hours.
For the milkshake: Combine the milk and 1/2 cup strawberry base in the blender and blend for 5 seconds. Add the ice cream and blend until smooth, about 10 seconds. Top with more strawberry base, the Sugar Biscuit and a strawberry for garnish, if using.
Preheat the oven to 450 degrees F.
Combine the flour, baking powder, baking soda and salt in a large bowl. Using a dough cutter, combine until the ingredients are incorporated. Scatter the cold butter pieces over the top of the flour mixture and continue to combine until the mixture resembles coarse meal. Add the buttermilk and mix just until the dough begins to come together.
Scrape the dough onto a lightly floured counter. Pat the dough into a 10 by 12-inch rectangle about 3/4-inch thick. Use a 3-inch round cutter to cut out the biscuits. Press together the scraps of dough and repeat the process. Place the biscuits on a baking sheet lined with parchment paper; brush the tops with cream and sprinkle with sugar. Bake the biscuits until lightly golden brown, 12 to 15 minutes. Remove from the oven and brush the tops with the remaining melted butter. Yield: 8 biscuits.