Cream sugar and 1/2 cup butter in the bowl of a stand mixer until fluffy. Beat in eggs, one at a time, and mix 1 1/2 teaspoons vanilla extract into mixture with the second egg.
Beat in flour and baking powder until thoroughly combined; beat in milk, lemon juice, and lemon zest to make a smooth batter.
Spoon the batter into the prepared muffin cups.
Bake in the preheated oven until cupcake edges are slightly golden brown and a toothpick inserted into the center of a cupcake comes out clean, 20 to 25 minutes. Cool cupcakes completely.
Beat 1 cup butter with 1 teaspoon vanilla extract and salt until smooth and creamy. Beat in confectioners' sugar, 1 cup at a time, to make a creamy frosting. Beat in blackberry jam. Spread frosting on cooled cupcakes.