A bran-enriched coarse wheat fraction containing bran, germ and starch is ground or milled to reduce grittiness without substantially damaging the starch because of abrasion to make whole wheat flour. The fraction may be stabilized by heating to reduce its lipase and lipoxygenase activity without gelatinizing the starch. During stabilization, high levels of natural antioxidants and vitamins are retained, yet acrylamide formation is substantially avoided. The stabilized coarse fraction or bran may be combined with a fine wheat fraction that contains predominantly endosperm to yield a stabilized whole wheat flour that exhibits extended shelf life and may be used for baking foods such as cookies with desirable oven spread.