In a large bowl, stir together ¾ cup flour, 1-⅔ cup water, and yeast. Let sit 45 minutes (to give yeast a head start on rising and to yield a richer flavor). Add remaining 3 cups flour, salt, sugar, olive oil, and 2 eggs. Stir until dough comes together; if it doesn't, add up to ½ cup more flour. Turn dough out onto a well-floured work surface and knead until a smooth, supple dough forms, 10-15 minutes. Clean the mixing bowl; then coat with oil. Cover with a kitchen towel and let rise until doubled in size, about 1-½ hours.

Turn dough back out onto a lightly floured baking sheet and fold over once or twice to deflate. Divide dough into three pieces. Weigh each piece to make sure they're equal; each one should weigh 14 ounces. Roll each piece out to a 16-inch-long strand; then pinch strands together at the top.

Braid strands together (like braiding hair), folding the right outer strand over the center one, then the left outer strand back over the center one as well. Repeat until loaf is fully braided; then pinch strands together at the bottom.

Whisk remaining egg with 1 tablespoon water to make a wash. Brush loaf with mixture, reserving some for a second wash after rising. Cover loaf with a lightly oiled piece of plastic wrap and let rise until doubled in size, 45 minutes to 1 hour.

Preheat oven to 375°. Brush loaf with remaining egg wash, sprinkle with poppy seeds if you like, and bake until challah is nicely browned, 35-45 minutes. Let cool before serving.