Preheat your oven to 375°F (190°C) and prepare two baking sheets by lining them with parchment paper. Combine the flour, cornstarch, and salt in a small bowl, whisking thoroughly to distribute the ingredients evenly. Set this mixture aside, and ensure all other ingredients are measured and at hand before proceeding.

Using a large bowl and a mixer, cream together the softened butter, brown sugar, and 1/3 cup granulated sugar until the mixture becomes pale, fluffy, and light in texture—this should take about 2 to 3 minutes. Add the vanilla extract and room-temperature milk, continuing to beat until everything is well blended and smooth.

Gradually add the dry mixture to the butter mixture, stirring gently until the ingredients are just incorporated—be careful not to overwork the dough. Fold in the mini chocolate chips and lemon zest, distributing them throughout. Let the dough sit undisturbed for about 5 minutes to allow it to firm slightly, making it easier to handle.

Portion the dough into 1-inch balls. Place the 3 tablespoons of granulated sugar in a shallow dish, then roll each ball through the sugar until it's completely coated. Transfer the sugar-coated balls to the prepared baking sheets, leaving approximately 1.5 inches of space between each one.

Use your thumb or the back of a teaspoon to gently press an indentation into the center of each dough ball, making a shallow well but being careful not to push through to the pan beneath. Spoon about 1 teaspoon of black raspberry jam into each well, pressing down lightly to secure it in place.

Bake in the preheated 375°F (190°C) oven for 13 to 15 minutes, watching for the edges to turn a light golden color and the jam to bubble gently. The centers may appear slightly underdone, but they will set as they cool. Remove from the oven and let the cookies rest on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.