Combine 1/2 cup warm water and yeast in a large mixer bowl; let stand 5 minutes to soften yeast. Heat 3/4 cup water and butter until warm (100° to 110°F). Add to yeast mixture along with 2 cups flour and remaining dough ingredients. Beat for 2 minutes at medium speed. Gradually add remaining flour until dough begins to pull away from the sides of the bowl. Turn out onto a lightly floured surface and knead for 7 to 9 minutes until smooth and elastic. Place dough in a greased bowl, turning once to coat. Cover and place in a warm, draft-free place and let rise for 1 hour or until double.
For filling, combine water, pineapple, apricots and sugar in a saucepan. Cook over medium-high heat until liquid is almost evaporated, 15 to 20 minutes, stirring occasionally. Stir in lemon peel. Set aside to cool.
Punch down dough; turn out onto lightly floured surface. Roll into a 10 x 15-inch rectangle. Place on greased or parchment lined baking sheet. Spread filling down center third of the rectangle. Cut 1-inch wide strips from outer edge to filling, on each side (using a sharp knife). Fold 1-inch of dough at each end over the filling, then continue crossing the strips alternately across filling to form braid. Whisk together egg white and water and brush over the top of loaf. Sprinkle with coarse sugar, if desired.
Let rise in a warm, draft-free place for 30 minutes or until nearly double. Bake in a preheated 350°F oven for 40 to 45 minutes. Loosely cover with foil if braid is getting too brown. Cool on a wire rack.