In a 2-cup liquid measuring cup, dissolve 1 teaspoon sugar in 1 cup warm water. Sprinkle yeast on top and let stand for about 5 minutes, until yeast gets all bubbly.
Make the dough in a stand mixer: Place 140 grams all-purpose flour and honey in the stand mixer bowl, fitted with the paddle attachment. Mix briefly to combine. Pour dissolved yeast mixture into machine and mix for about 1 minute. Add egg and oil and mix again for about 30 seconds. Add remaining 285 grams of all-purpose flour, whole wheat flour and salt and mix on medium-low speed for about 1 minute. Switch to dough hook and knead for 8-9 minutes.
Turn dough out onto a lightly floured surface. Knead for a minute and then form the dough into a ball and transfer it to a large oiled mixing bowl. Turn dough until all sides are coated. Cover the bowl with plastic wrap or a damp towel. Let it rise at room temperature for about 1½-2 hours, or until doubled in bulk. A good test to see if the dough has risen enough is to insert your finger to a depth of 1 inch and wait to see if the impression remains. If the impression fills in, wait another 15 to 20 minutes to let rising continue.
While dough is rising, prepare topping. Place ingredients in a food processor and pulse several times until crumbly. Store in fridge until needed.
After the dough has risen, punch it down to deflate it. Divide dough it into 10 equal pieces (Do yourself a favour and weigh dough and divide by 10 to be precise). Roll each piece into a ball. Join the last 2 balls together to make one larger ball. You will have 8 smaller balls and one larger ball, for a total of 9 balls of dough. Place balls on a parchment lined baking sheet. Cover with a towel and let rest for about 10 minutes.
Roll each of the smaller balls into a 14-inch-long strand with slightly tapered ends. Roll the large ball into a 21-inch strand. Lightly dust each strand in flour and place on parchment lined baking sheet so they do not stick.
Beat eggs with water and place in a shallow baking dish. Place crumble topping in a second shallow baking dish. Dip 4 strands in egg and then in the crumble topping. Gently press topping into dough so that it sticks well. Coat the larger strand in crumble topping as well. You will have 4 plain strands and 5 crumble coated strands.
Line a baking sheet with parchment paper. Preheat oven to 350°F. Watch this how to video on my blog for shaping instructions and form the challah. Cover the loaf with a tea towel and let rise at room temperature for 45 minutes to 1 hour. Beat egg yolk with water. Using a thin brush, brush only the plain strands with the egg wash. Add a bit more crumble topping to the crumb coated strands. Bake for 20 minutes. Remove from oven, and on the parts where the bread has expanded, paint any newly exposed dough with egg wash on the plain strands and fill in with more crumble topping on the expanded bits of crumb coated strands. Place challah back in the oven and bake for a further 20-25 minutes until golden brown, or until an instant read thermometer, inserted into the center of the loaf, registers 190°F. Remove challah to a wire rack and let cool before serving.