Set the oven to 400°F and prepare foil, a small baking dish, and a larger dish that can hold it. The smaller dish needs room around its sides for the water bath.

Empty the condensed milk into the small dish, cover with foil, and nest it inside the larger dish with water reaching halfway up its sides. Bake for 1 hour to 1 hour and 15 minutes, checking the water at 45 minutes and replenishing it if evaporation has lowered the level.

Take the baking dishes out of the oven and leave them undisturbed for about 15 minutes. This pause lets the hot dishes and filling cool enough for easier handling.

Uncover the inner dish, lift it from the water, remove the cinnamon stick if present, and transfer the filling to a glass or stainless steel bowl. Whisk until the caramel looks uniform and smooth.

Keep prepared manjar blanco refrigerated in a tightly covered container if assembling later. To soften chilled filling, stand the closed container in warm water until the caramel becomes spreadable.

Combine the flour, sifted cornstarch, sifted powdered sugar, and salt in a medium bowl with a whisk. The dry mixture should look evenly blended, without pockets of starch or sugar.

Beat the butter with the vanilla, if using, in a stand mixer or with an electric hand mixer, then briefly mix in the optional yolk. Stop when the butter mixture looks smooth and the yolk, if included, has disappeared into it.

Add the dry mixture and run the mixer slowly for about 1 minute, stopping as soon as the dough comes together. Low speed keeps the powders in the bowl and avoids unnecessary mixing.

Gather the dough into a ball, press it into an approximately 2-inch round on plastic wrap, and wrap tightly before refrigerating for about 1 hour. The chilled dough should feel firmer and less sticky than it did immediately after mixing.

Set the oven to 350°F before rolling the rested dough. This lower temperature is for the cookies, not the caramel.

Roll the dough slowly between two sheets of parchment to an even thickness within the 1/2- to 1/4-inch range. If it resists rolling or cracks from the cold, allow about 10 minutes at room temperature before continuing.

Cut closely spaced rounds with a 3-inch cutter and lift them with an offset spatula onto a parchment-lined baking sheet, leaving about 2 inches between cookies. Chill each sheet of cutouts for about 10 minutes, then continue with the remaining dough and reroll the scraps about twice as needed.

Lightly mark the tops with fork tines, then bake for about 11 to 13 minutes and leave the cookies on their sheets until they reach room temperature. Remove them when the tops feel lightly firm and set while the edges remain completely unbrowned.

Arrange half the fully cooled cookies bottom-side up and, if using a piping bag with a tip, pipe a round of manjar blanco onto each one. The filling should flow smoothly without requiring pressure against the cookie.

Use a spoon instead of piping if preferred, gently leveling the filling with a butter knife. Support the cookie and keep the knife pressure light to protect the delicate disk.

Place a second cookie over each filled half, press very lightly, and roll the exposed sides in sifted powdered sugar. Stop pressing once the halves hold together rather than forcing the caramel outward.

Transfer the finished sandwiches to an airtight container or sealable bag. A closed container protects their delicate crumb from drying out.