Mix active and quick-rise yeast with 3 tablespoons water in a bowl to make a paste. Mix 4 cups of flour with 1 cup water in a large bowl until thoroughly combined; stir yeast mixture into flour mixture.
Beat remaining 1 cup water, baking powder, salt, sharp Cheddar cheese, habanero pepper, and rosemary into dough; beat remaining 1 cup flour into dough. Turn out onto a floured work surface and knead until dough is soft and elastic, at least 6 minutes. Form dough into a ball, place into an oiled bowl, and turn the dough around in the bowl to pick up a thin coating of oil. Cover bowl and let rise for 2 hours.
Punch the dough down, turn out onto a floured work surface, and knead for 6 more minutes. Divide dough in half and form each half into a loaf; place loaves on a baking sheet. Cover loaves and let rise until doubled, 15 to 20 minutes.
Preheat oven to 400 degrees F (205 degrees C).
Bake loaves in the preheated oven for 15 minutes. Reduce oven heat to 350 degrees F (175 degrees C) and bake until golden brown, 20 to 25 minutes.