In a separate small bowl, whisk together the egg, egg yolk, milk, and vanilla. Add this to the flour-butter mixture and mix on low speed until the dough comes together in large clumps, about 30 seconds. Transfer the dough to a work surface and knead briefly to form a cohesive mass. Divide the dough in half (about 490g per piece).
Shape the two portions of dough into rough 5" x 7" rectangles. Wrap the dough tightly in plastic wrap or your favorite reusable wrap and refrigerate for at least 20 minutes and up to 5 days. (If the dough is left to chill for longer than 20 minutes, let it warm up at room temperature for about 30 minutes before rolling.)
To make the filling: While the dough is chilling, combine all the filling ingredients except the egg white in the mixing bowl; mix on low speed, scraping the bowl as needed, until it’s uniform in color and the texture of wet sand.
To assemble: Once the dough has chilled, line a baking sheet with parchment. Remove the dough from the refrigerator and lightly flour your work surface.
Working with one piece of dough at a time (keep the other piece wrapped and refrigerated), roll the dough into an 11" x 13" rectangle measuring 1/8" thick. As needed, trim irregular edges and use the trimmed pieces to patch any cracks and fill out the rectangle. (Bake any extra scraps separately as a baker’s snack!) Use additional flour sparingly to ensure the dough does not stick to the work surface.
Use a bench knife or spatula to pick up one short edge of the rectangle and drape it onto the rolling pin. Roll the whole rectangle of dough gently around the rolling pin, being careful not to compress it in the process.
Unfurl the dough onto the prepared baking sheet by unrolling it off the pin. Refrigerate the dough, uncovered.
Working as quickly as possible, roll the remaining piece of dough into an 11" x 13" rectangle measuring 1/8" thick. Use the same process to trim irregular edges and patch any cracks.
Remove the chilled dough from the refrigerator and brush a 1" border of the egg white mixture on all edges of the dough.
Sprinkle the filling evenly over the dough, breaking up any large clumps as you go, leaving the 1" border of egg washed dough bare. Lightly press the filling into an even layer.
Use a bench knife or spatula to pick up one short edge of the second 11" x 13" dough rectangle and drape it onto the rolling pin. Roll the whole rectangle of dough gently around the rolling pin, being careful not to compress it in the process.
Unfurl the dough on top of the filling, lining up the edges of the top piece of dough with the bottom piece as best you can. Gently press out any large air pockets and then use the tines of a fork to seal the edges. Refrigerate the assembled pop tart, uncovered, for 30 minutes. (Or wrap it with plastic wrap or your favorite reusable wrap and refrigerate for up to 24 hours.)
While the pop tart chills, preheat the oven to 350°F with a rack in the center.
When ready to bake, use the tines of a fork to poke the surface of the pop tart about two dozen times, creating steam vents.
Bake for 30 to 35 minutes, or until the edges are light golden brown; the surface may be slightly shiny and/or puffy. Let it cool on the pan for 5 minutes while you make the icing.
To make the icing: In a medium bowl or the bowl of a stand mixer fitted with the flat beater, mix the icing ingredients together. Using a small offset spatula or a butter knife, spread the icing evenly over the top of the giant pop tart, leaving a 1" border bare. Let the icing set for 15 minutes before slicing and serving.
Storage information: Store any homemade pop tart leftovers, well-wrapped, at room temperature for up to 2 days. Reheat individual portions in the microwave as desired.
