Place 1/4 cup water in a bowl. Sprinkle in yeast; stir until dissolved. Add remaining water, sugar, salt, caraway seed, butter, rye flour, and 1 / 2 cup all-purpose flour. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Stir in enough remaining flour to make a soft dough. Knead on lightly floured surface until smooth and elastic, about 8 to 10 minutes. Place in greased bowl, turning to grease top. Cover; let rise in warm, draft-free place until doubled in size, about 1 hour.
Punch dough down; divide in half. Cut each half into 12 equal pieces. Roll each piece into a rope about 6 inches long. Place on greased baking sheets, about 2 inches apart. Cover; let rise in warm, draft-free place until doubled in size, about 30 minutes.
Brush lightly with melted butter; sprinkle with caraway seed. Bake at 400°F for 15 to 20 minutes or until done. Remove from baking sheets; cool on wire racks.