|Given a monoid S with E any non-empty subset of its idempotents, we present a novel one-sided version of idempotent completion we call left E-completion. In general, the construction yields a one-sided variant of a small category called a constellation by Gould and Hollings. Under certain conditions, this constellation is inductive, meaning that its partial multiplication may be extended to give a left restriction semigroup, a type of unary semigroup whose unary operation models domain. We study the properties of those pairs for which this happens, and characterise those left restriction semigroups that arise as such left E-completions of their submonoid of elements having domain 1. As first applications, we decompose the left restriction semigroup of partial functions on the set X and the right restriction semigroup of left total partitions on X as left and right E-completions respectively of the transformation semigroup on X, and decompose the left restriction semigroup of binary relations on X under demonic composition as a left E-completion of the left-total binary relations. In many cases, including these three examples, the construction embeds in a semigroup Zappa-Szép product.