In a large mixing bowl, stir yeast into cranberry or beet juice. Add oil, honey, and salt. Add 1 cup of the flour. Beat with an electric mixer on low speed for 30 seconds, scraping the bowl. Beat on high speed 3 minutes. Using spoon, stir in as much remaining flour as you can.
Turn out onto a lightly floured surface. Knead in enough remaining flour to make a moderately stiff dough thats smooth and elastic (6 to 8 minutes).
Place the dough in a lightly greased bowl. Turn the dough once to grease the surface. Cover the dough and let it rise in a warm place until doubled (about 1 hour). Punch the dough down. Cover and let it rest for 10 minutes.
Shape into 12 smooth balls. Place balls in greased muffin cups. Cover; let rise in a warm place for about 30 minutes or until nearly doubled.
Bake in 375 degree F oven 12 to 15 minutes or until brown. Serve warm. Makes 12 rolls.
To get 1 cup of beet juice, drain the juice from a 16-ounce can of beets. Add enough water to make 1 cup.