[Oden]

How To Buy and Restore a Classic Car by Terry Bolton

That is the absolutely best way to do it.

The Best way to increase traffic to your site

So what is the absolute best way to drive traffic to your site?

Are both grammatically correct?

[Oden]

[susan53]

Either.

absolutely

This is OK, but it would be more usual to say:That is absolutely the best way to do it.
than to use the word order you've suggested. I did a Google battle and got :
absolutely the best - 4,520,000 examples
the absolutely best - 1,650,000 examples

absolute

But the absolute best brought up 13,300,000 examples. So in wins out over both the others. To me it sounds very informal and emphatic however. So I'd suggest ...

in neutral style - absolutely the best is usual
more informally - the absolute best is fine

[Oden]

[clevermae]

For me absolutely means you are conducting a form of expression, while absolute means an exact word to point something that you think is perfect!