Heat vegetable oil in a skillet over medium-high heat. Add onion and bell peppers; cook 3 minutes, stirring frequently. Remove from heat; let cool.
Combine 1 cup flour, sugar, undissolved yeast and salt in a large mixing bowl. Heat water, milk and shortening until very warm (120° to 130°F). Gradually add to flour mixture. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Stir in cooked onion mixture, cheese and enough remaining flour to make a soft dough. Knead on lightly floured surface until smooth and elastic, about 8 to 10 minutes. Cover; let rise 10 minutes.
Roll dough to 18 x 10-inch rectangle; sprinkle with bacon bits. Roll up tightly from long side as for jelly roll; pinch seam to seal. Place in a greased 9 or 10-inch tube pan. Join ends; pinch to seal. Cover; let rise in warm, draft-free place until doubled in size, about 1 hour.
Bake at 375°F for 35 to 40 minutes or until done. Let cool slightly in pan on wire rack for 5 minutes. Remove from pan; serve warm.