2cupssourdough starter (460g)(preferably active, but this bread can work with a new starter as well as a discard, just with slightly less rise)
1cupwater (230g)
1cuplukewarm milk (230g)
1tablespoonsolive oil (14g)
1tablespoonshoney (23g)(or sugar)
2 ½cupsunbleached flour (345g)
2 ¼cupswhole wheat flour (325g)(if you can find white whole wheat, it's a bit lighter texture)
1tablespoonsalt (17g)
1 ½cupsdiced sharp cheddar cheese* (200g)(or grated cheese, if you prefer)
Instructions
The morning of the day you'd like to bake, add to a mixer bowl the active (fed the night before) sourdough starter, water, milk, oil, honey, and flours. Mix for about 2 minutes until everything is combined. Let sit for 15 minutes. (UPDATE: this is for an autolyse, which I don't do anymore - if you'd like you can simply mix all the above ingredients with the salt and then move to the next step right after mixing all together for a few minutes.)
Add the salt (if you haven't) and mix with a dough hook for another 4-5 minutes. Cover the mixer bowl with plastic or something airtight and let rise until doubled, about 4 hours.
When the dough has doubled, add the cheese and mix with the hook until distributed through the dough.
Divide between two greased 8.5 x 4.5-inch loaf pans, pressing and smoothing the tops as needed (the pans should be filled halfway with the dough). Oil the tops and cover with plastic again to rise for 2 more hours or until the dough is at (or almost at) the top of the pans.
Heat the oven to 400 degrees. When hot, bake the loaves for 30-40 minutes, turning as needed for even cooking, until browned and internal temperature is about 200 degrees.
Cool for a few minutes in the pans before turning out onto a cooling rack. Serve warm or toasted.
Notes
*Other medium-hard cheeses work, too, like Fontina, Muenster, or Romano. Some commenters have really liked the bread with pepper jack, as well