Can one start a sentence with "as well" ?

Short answer: For the most part, yes, you can.

There are no grammar rules that restrict or prohibit the use of “as well” at the beginning of a sentence. However, more conservative and traditional linguistics tend to avoid the use of ‘as well’ at the beginning of a sentence because it makes for an awkward sentence.  

The use of “as well” at the start of a sentence may not be the best choice in certain sentences. Using “as well” at the start of sentence is considered interchangeable with “furthermore” and ‘also’, as long  as it is followed by a complete sentence and/or a verb so that it can stand on its own.

     Use of “as well” as a comparative phrase:

o   Correct: As well as his teacher, he speaks Spanish fluently.

o   Incorrect: He spoke Spanish fluently. As well as his teacher.

  Use of “as well” with a verb in a sentence:

o   Correct: He can play piano. As well, he can play violin.

o   Incorrect: He can play piano. As well, violin.

Take home lesson: use ‘as well’ to start a sentence, but make it a complete sentence. However, it is considered awkward or unacceptable in formal or professional writings per grammar purists.

At the end of the day, language is an art. It is fluid, dynamic and it lends itself to creative expressions.

        “The limits of my language is the limits of my world” –Ludwig Wittgenstein.