In the bowl of a stand mixer fitted with the whisk attachment, whip the cold heavy cream on medium-high speed until stiff peaks form, about 3 to 4 minutes. Switch to a hand whisk or a spatula, and gently fold in the sweetened condensed milk and vanilla pudding powder until fully combined.
Line a 9 x 13-inch/23 x 33 cm baking pan with parchment paper. Pour in the ice cream mixture and spread into an even layer. Freeze for at least 6 hours or overnight until solid.
In a large mixing bowl, using an electric hand mixer or a stand mixer fitted with the paddle attachment, cream the butter and powdered sugar together until smooth and fluffy, about 2 minutes. Add the egg yolks one at a time, beating well after each addition. Mix in the vanilla extract. Add the cornstarch, flour, and baking powder. Mix just until a soft dough forms. Wrap the dough in plastic wrap and refrigerate for 30 minutes.
Place the chilled dough between two sheets of parchment paper and roll it out to a thickness of about 3.5 mm (slightly thicker than 1/8 inch). Using a 6 cm/2.5-inch round cookie cutter, cut out circles and transfer them to a parchment-lined baking sheet, spacing them slightly apart.
Preheat the oven to 350°F (180°C), using top and bottom heat if available. Bake the cookies for 10 to 12 minutes, until set but still pale. Let the cookies cool completely on the baking sheet.
Remove the frozen ice cream from the freezer. Using the same 6 cm/2.5-inch cookie cutter, cut out rounds of ice cream. With the back of a teaspoon, gently press a small indentation in the center of each ice cream round and fill it with about 1 teaspoon of dulce de leche. Sandwich the ice cream between two alfajores cookies. Insert a wooden popsicle stick into the center of each sandwich. Place the assembled sandwiches on a parchment-lined tray and return them to the freezer for 1 hour to firm up.
In a microwave-safe bowl, melt the milk chocolate in short 20-second bursts, stirring between each burst, until fully melted. Stir in the neutral oil until smooth. Dip each ice cream sandwich into the melted chocolate, letting the excess drip off. Immediately sprinkle shredded coconut around the sides. Place the finished sandwiches back on a parchment-lined tray and freeze until ready to serve.