Combine the whole wheat flour, all-purpose flour, yeast, sea salt, water, 2 of the eggs, olive oil, and honey in the bowl of a standing mixer fitted with a dough hook. Give the mixture a few turns with a rubber spatula to moisten all of the ingredients, and then knead on medium-high speed until your dough is smooth, 4 to 6 minutes. It will cling to the hook and clear the sides of the bowl. Add the apricots and knead until they’re just incorporated, another minute or two.
Transfer the dough to a lightly oiled bowl or dough-rising container, cover the bowl with plastic wrap, and let stand at room temperature until it has doubled in volume, 1-1⁄2 to 2 hours.
Gently press on the dough while it’s still in the bowl to deflate it, and then turn it onto a lightly floured countertop. Divide it into 3 equal portions and, with the palms of your hands, roll each portion into a 15-inch-long rope, just the way you used to roll clay into ropes when you were a kid.
Line a baking sheet with parchment paper and place the 3 dough ropes side by side on the sheet. Pinch the ropes together at one end, tucking them under the loaf. Braid the ropes together, right over center and then left over center, as tightly as you can, until the ropes are too short to braid. Pinch the ends of the braid together and tuck them under the loaf as you did with the other end.
Dust the loaf lightly with flour, drape with plastic wrap, and let rise at room temperature until almost doubled in volume, 1 to 2 hours (alternatively, refrigerate the covered loaf overnight and bring it to room temperature before letting it rise completely).
Heat the oven to 350°F. Lightly beat the remaining egg and brush it all over the top of the loaf. Bake the challah until the top is deep golden brown and the bottom makes a hollow sound when tapped, about 40 minutes. Transfer the bread to a wire rack to cool completely before slicing and serving. Whole Wheat Challah with Apricots will keep at room temperature in an airtight container for up to 2 days. For longer storage, freeze in a zipper-lock plastic bag for up to 1 month. To defrost, place on the countertop for an hour or two, and reheat in the oven at 350°F for 5 minutes before serving.
For whole wheat challah with green olives, replace the apricots with 1 Tbs. finely chopped pitted green olives (any green olive will work here, but I love big, juicy Cerignolas) for a savory version of whole wheat challah, and serve with roast chicken or brisket.