Bluetooth handsfree better quality

Short:

No

Long:

There are so called Profiles in Bluetooth, and only one can be active at a time.

A2DP supports one-way HQ audio.

HSP/HFP support two-way audio, but only with very poor quality (HFP 1.6 added mSBC, which is a 16kHz mono codec, before, it was even worse).

(ref http://www.tomshardware.co.uk/answers/id-3399361/bluetooth-headsets-high-quality-audio-microphone-simultaneously.html)

(I added the same answer to both Is it possible to use a bluetooth headset mic while having High quality sound and Bluetooth handsfree better quality because they are found using google.)

It is even more complicated: not only HFP (Hands Free Profile) 1.6 (2011), but even HFP 1.7 (2018) consider mSBC codec (the prerequisite for feature called Wide Band Speech - basically sample rate: 16 kHz / 16bit) optional in both

• HF (Hands Free - the headphones)
• AG (Audio Gateway - the computer, cellphone).

See Table 3.1, page 19.

Also:

Since it is only the AG that knows if wide band speech should be used, it should always be the AG that establishes the Synchronous Connection with the required codec.

2.3 d, page 16.

So it is not uncommon that the same HF working perfectly with mSBC under Android AG (e.g. Jabra Elite 65t) fails miserably under Windows 10 with mSBC and all you can do is buying Jabra Evolve 65t for 3x higher price to get the expected result. Or replace Windows with Android...

Oh, you thought of even better quality than mSBC/WBS? Unfortunately, no chance unless somebody comes with "A2DP microphone" :-(