Warm the milk on the stovetop or in the microwave on a low setting. You just want to get it warm, no more than 110°F/43°C.
Add the warm milk and yeast to a large bowl (the stand mixer bowl, if you're using one) and allow to sit, covered, for 5 minutes.
Add the flour, sugar, salt, butter and egg to the bowl with the milk and yeast. Either with the paddle attachment of a stand mixer or a very sturdy spoon, mix together until a sticky dough forms.
Add the cranberries, apples, zest, cinnamon and vanilla extract to the bowl.
If you're using a stand mixer, mix again until the dough starts pulling away from the side of the bowl, 2 to 5 minutes. If you are going to knead by hand, dump the contents of the bowl onto a floured surface. Knead well, about 5 minutes, until the dough becomes smooth and stretchy.
Proofing and Shaping
Lightly oil a large bowl (keep in mind that the dough will at least double in size). Place the kneaded dough in the bowl and turn it to lightly coat all sides of the dough in oil.
Tightly cover the bowl and let it rise for about 1 hour, or until doubled in size. If you are in cold kitchen, it will take longer, or shorter if your kitchen is warm.
Line a 9"x13" baking tray with parchment paper.
Once the dough is doubled in size, it's time to shape the buns. On a floured surface, divide the dough into 12 roughly equal pieces. (If you want to be crazy like me, weigh the entire dough, divide the weight by 12 and you can weigh out each piece.)
Roll each piece into a ball, pinching it at the bottom to seal the dough. Place the balls pinch-side down in your baking tray, giving each one some room to expand.
Cover the baking tray and let the buns rise for another hour.
Make the Crosses and Bake
While the buns rise, preheat the oven to 400°F/205°C/gas mark 6.
In a medium bowl, mix the flour with the water until it forms a thick but pipeable paste. Place the paste in a piping bag fitted with a round tip (or just cut an even hole in the bottom of a piping bag).
Pipe a line across the center of all of the buns, and do that for each row of buns. Then pipe a line across all of the buns in the other direction.
Place the baking tray in the oven and bake for 20-25 minutes, or until golden brown.
Glaze
On the stovetop, heat the jam on a low temperature just to melt it. You can pick out any large chunks, or pass it through a strainer to get all of the chunks out.
Using a pastry brush, brush the warm jam onto the warm buns. Serve the buns warm. They will keep for a couple of days at room temperature, or freeze individually wrapped buns for best freshness.
Notes
If you prefer to pipe an icing cross instead of the flour paste, skip the flour paste steps. After the buns have cooled, mix together a cup (120 g) of confectioners' sugar, 1 teaspoon of vanilla extract and about 3 tablespoon milk or cream. Add more sugar or milk to get the right consistency for a thick, but pipeable frosting, and pipe the crosses with that.