In large bowl, combine 2 cups bread flour, sugar, undissolved yeast, caraway seed and salt. Gradually add very warm water and oil to dry ingredients; beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Add rye flour gradually for an additional 2 minutes, scraping bowl occasionally.
Knead on lightly floured surface (or in mixer at medium speed) until smooth and elastic, about 6 to 8 minutes. Place in greased bowl, turning to grease top. Cover; let rise in warm, draft-free place until doubled in size, about 30 to 45 minutes.
Punch dough down. Remove dough to lightly floured surface; Form into 5-inch ball. Place on large greased baking sheet or parchment paper. Cover; let rise in warm, draft-free place until doubled in size, about 30 to 45 minutes. If not baking right away place in refrigerator; remove from fridge 30 minutes prior to baking.
With sharp knife, make 4 slashes (1/4-inch deep) in crisscross fashion on top of each loaf. Brush with egg white mixture. Bake at 400ºF for 35 minutes or until done; cover with foil halfway through baking time to prevent excess browning. Remove from sheet; cool on wire rack
ALTERNATIVE BAKING METHOD (the way we cooked the photographed loaf) Oven at 450F with pizza stone in oven. Form loaf on cookie sheet covered with parchment paper and let rise for 30 -45 mins (this is step 3 sse above), apply egg wash, slit loaf, trim excess parchment paper. Slide loaf onto hot stone and cover with an disposable aluminum roasting pan. Bake for 35 mins. With this method you will produce a loaf with a crustier crust than just baking in the oven without the pan over it!