In a large bowl, dissolve yeast in warm water. Add the milk, sugar, butter, eggs and salt; mix well. Stir in enough flour to form a soft dough. Do not knead. Place in a greased bowl, turning once to grease top. Cover and let rise in a warm place until doubled, about 1 hour.
Punch dough down. Turn onto a lightly floured surface; roll into a 15x8-in. rectangle. Cut into 3x1-in. rectangles. Place on greased baking sheets. Cover and let rise in a warm place until doubled, about 30 minutes.
In an electric skillet or deep-fat fryer, heat oil to 375°. Fry, a few at a time, until golden brown on both sides. Drain on paper towels.
For filling, in a small saucepan, combine flour and milk until smooth. Bring to a boil. Cook and stir for 2 minutes or until thickened and bubbly; cool.
In a small bowl, cream butter and sugar until light and fluffy. Beat in vanilla. Add cooled flour mixture; beat until fluffy. Gradually beat in enough confectioners' sugar to achieve a spreadable consistency.
While sticks are still warm, make a cut in the top of each; fill with filling.
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