Combine water and sugar in a small bowl. Sprinkle the yeast over the top. Set aside until foamy, about 8 minutes.
Put 1 cup of the flour in a large bowl and make a well in the center. Whisk honey with 1 of the whole eggs, the yolks, olive oil, and salt in a small bowl and pour into the well. Stir to combine. Add yeast mixture, remaining 3 cups flour, and currants and stir to make a soft, shaggy dough. Knead dough on a lightly floured surface until soft and supple, about 10 minutes.
Shape dough into a ball and place in lightly oiled large bowl. Cover with a clean kitchen towel and let rise until doubled in size, about 1 hour. Turn dough onto a lightly floured surface; knead for just a minute, shape into a ball and return to bowl. Cover and let rise again until doubled in size, about 1 hour.
Line a baking sheet with parchment paper. Form loaf according to the following instructions: Turn dough onto a clean work surface and divide into 3 equal pieces. Roll and stretch each piece into a rope 12 to 15 inches long. Arrange ropes side by side on the work surface. Starting at the far end, braid the ropes. When the braid is finished, tuck both ends under and pinch to seal. Place loaf on prepared baking sheet.
Cover with a clean, dry kitchen towel and set aside to rise until doubled, about 1 hour.
Place a rack in the center of the oven and preheat to 375 degrees F. Beat remaining egg. Brush loaf with egg and sprinkle with poppy seeds. Bake until golden brown, about 35 minutes.
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