Cereals and oilseed flours increase the nutritional value of pastry products. Their composition is different from traditional wheat flour, which means their technological properties are also different. The research objective was to study the functional and technological properties of various types of flour to develop a multicomponent powder mix formulation for functional cookies. The research included baking wheat flour of the highest grade, whole-wheat flour, corn flour, semi-skimmed flax flour, and rice flour. The flours and their multicomponent mix were tested for functional and technological properties, i.e., water-holding capacity, fat-retaining capacity, fat-emulsifying capacity, emulsion stability, etc. These propertied appeared to depend on the protein and carbohydrate composition of the flour. The water-holding capacity of the flax flour sample was 6–8 times greater than that of other types, while the fat-retaining capacity of the corn flour sample was 30% higher. Corn and flax flours demonstrated the best fat-emulsifying capacity and emulsion stability. The research resulted in a new multicomponent corn-flour-based powder mix for high-quality butter cookies. The formulation included corn flour and vegetable oil, which increased the nutritional value of the final product and expanded the range of gluten-free cookies.