Maltosyltransferase that uses maltose 1-phosphate (M1P) as the sugar donor to elongate linear or branched alpha-(1->4)-glucans. Maltooligosaccharides with a degree of polymerization (DP) superior or equal to 4 are efficient acceptors, with DP6 being optimal in the GlgE-catalyzed polymerization with M1P. Is probably 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.