"Especially" and commas

The rule you were taught is an over-generalization. The sentence written above is correct with no commas at all. The "rule" about especially only applies to the case when especially is being used as a parenthetical phrase, usually at the end of a sentence.

We really like ice cream, especially chocolate ice cream.

This rule is not unique to the word especially, but is used for all sorts of sentence-final parentheticals:

The "rule" about especially only applies to the case when especially is being used as a parenthetical phrase, usually at the end of a sentence.

Since your sentence is not using especially to begin a parenthetical statement, it does not require a comma. It is, however, stylistically awkward, and I would recommend the following revision:

Secondary electron emission from metallic surfaces, especially dielectric surfaces, is a fundamental process of great technological significance

If you put any commas in that sentence at all, then put one after metallic and another after dielectric.