Maltosyltransferase that uses maltose 1-phosphate (M1P) as the sugar donor to elongate linear or branched alpha-(1->4)-glucans. Is involved in a branched alpha-glucan biosynthetic pathway from trehalose, together with TreS, Mak and GlgB. [(1->4)-alpha-D-glucosyl](n) + alpha-maltose 1-phosphate = [(1->4)-alpha-D-glucosyl](n+2) + phosphate Homodimer. Belongs to the glycosyl hydrolase 13 family. GlgE subfamily.