Place the flour, sugar, and cold butter chunks in the bowl of a food processor. Pulse minimally until the butter is mostly incorporated and a loose, crumbly powder forms.

Add 6 tablespoons of water to the food processor and pulse again until clumps begin to form. If the clumps can be hand-formed into a dough at this point, move to step 3. If the mixture is too crumbly to form into a dough, more liquid is needed—add additional cold water 1 tablespoon at a time just until you can hand-form the clumps into a dough.

Form the dough into a rectangular patty, wrap it tightly in plastic wrap, and refrigerate for at least 1 hour.

Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper or a silicone mat.

Remove the dough from the fridge. Unwrap it and place on a large lightly floured surface or countertop. Using a floured rolling pin, gently roll the dough out to a 1/8-inch thickness, or as thinly as possible without breaking it. The total size should be at least 12½ × 16 inches. (If your work surface is not large enough, split the dough into two parts before rolling it out and refrigerate the second part until ready to use.)

Once the dough is rolled into a thin sheet, use a pizza cutter or sharp knife to cut out 2½ × 4-inch rectangles of dough. While rolling, press together broken edges of dough as needed to fit as many rectangles as possible. You should be able to get 20 to 24 rectangles out of the dough.

Using a thin spatula, place half of the dough rectangles on the prepared baking sheet. Fit as many as you can, leaving ½ inch between each slice. Add 1 to 2 tablespoons of jam along the center of each dough slice. Place an additional dough slice on top of each jam-covered slice and seal all four edges using a fork, gently pressing down the fork prongs into both layers of dough. If the dough doesn’t seem completely sealed, it’s okay. There is no harm if some jam seeps out during baking. Using a toothpick, pierce three or four holes on the top of each pastry. Repeat until you’ve used up all your dough.

Bake the pastries for 25 to 30 minutes, until very lightly golden around the edges. Remove from the oven and allow to cool for a few minutes on the baking sheet. Using a spatula, transfer the pastries to a cooling rack to completely cool before icing.

To prepare the icing, pour the powdered sugar and vanilla soy milk into a small bowl and whisk until combined. The icing should be thick so that the majority will stay atop the pastry.

Place a paper towel or parchment paper below the cooling rack to catch icing drips. Spoon the icing onto the middle of the cooled tarts, nudging it outward and allowing some to naturally drip off the sides. Add the sprinkles before the icing dries and hardens.