Dos palabras casi gemelas que confunden hasta a nativos. Especially significa sobre todo, en particular y sirve para destacar un caso dentro de un grupo: I love fruit, especially mangoes. Specially significa a propósito, para un fin concreto y suele acompañar a un participio: a cake specially made for you. Truco rápido: si puedes sustituirlo por "sobre todo", es especially.
Cuál va
ENespecially = above all: It's cold, especially at night.
ESespecially = sobre todo: It's cold, especially at night.
ENspecially = for a purpose: shoes specially designed for running.
ESspecially = para un fin: shoes specially designed for running.
ENespecially also goes before adjectives = particularly: It's especially busy today, not especially hot.
ESespecially también va ante adjetivos = particularmente: It's especially busy today, not especially hot.
El error típico del hispanohablante
Para "en especial / sobre todo" usa especially, no "specially": especially in summer.