In stand mixer bowl, add 1 cup bread flour, yeast, sugar, salt and caraway seeds; mix well. Add warm water, oil, and sourdough starter to flour mixture. Using a paddle attachment, blend on low speed for 30 seconds; beat on medium speed for 4 minutes.
Switch to dough hook attachment. Gradually stir in rye flour and enough remaining bread flour to make a firm dough. Knead until smooth and elastic, about 5 to 7 minutes.(NOTE: For hand mixing: following above steps, mix ingredients in large mixing bowl using a wooden spoon or dough whisk. Knead on floured surface until smooth and elastic.)
Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until indentation remains after poking dough with finger down to second knuckle, about 30 minutes.
Turn dough onto lightly floured surface; punch down to remove air bubbles. Shape into a round loaf and place on a greased baking sheet. Cover; let rise until indentation remains after lightly touching.
Preheat oven to 375°F. (See note below to add steam for a crusty loaf.)
With a very sharp knife or bread lame, make an “X” on the top of loaf. Bake for 30 to 40 minutes. Remove from baking sheet; cool on wire rack.
For a crusty loaf: Place a metal broiler tray on bottom oven rack before preheating oven. After placing loaf in oven to bake, quickly and carefully pour 1 cup hot water into broiler tray and close oven door. Bake as directed.