No, it doesn't work here.

"We are particularly dissatisfied with the technical support you gave us."

Or, if it fits:
In particular, we are not satisfied with the technical support you gave us. (or "...we received.")

I agree, sounds so much better!

Thank you!

I agree. I can't think of a case where a sentence would begin with "especially".

On the other hand, in less formal contexts, even written, it's not uncommon to have sentence fragments. Some people consider this incorrect, and you wouldn't find it in the New York Times. But in speaking or less formal writing, certainly.

"I told my grandmother not to drive without her glasses. Especially at night."