Which sounds better 2 way or 3 way speaker design

I agree with @knotscott - quality of the speaker design, parts and manufacturing matter most. But all things being equal (which is almost never true) I prefer a great 2-way design with CD horns to 3-way. I have owned a lot of great 3-way speakers and I can count on one hand the ones where I couldn't notice the effects of the xover. Some more, some less.

Even if you are limiting this discussion to conventional cone drivers (e.g., not considering electrostatic or planar magnetic panel speakers, bending wave drivers like MBL speakers), there are great examples of one, teo, three, and more-way speakers.  No speaker is perfect in all respects and each has different strengths and weaknesses and listeners have different tastes snd priorities such that a wide range of speakers can be favored.  I like the one-way, single driver systems from Songer Audio and Charney Audio.  I like MANY two-ways, some conventional driver systems from the likes of Audio Note (including their near six figure fieldcoil two-way AN-E speaker) ProAc, Harbeth, Fyne, etc., and MANY three ways, particularly three-way horns.  

I don’t think this particular design characteristic is determinative of sound quality.

I’m partial to my 2.5-ways. They use a JBL 18", a JBL 9.7", and a modified Heil. (BTW, a Heil is equivalent to 8 square inches of tweeter surface.) And these do keep the midrange in the 9.7 going all the way up to ~3,500Hz before crossing over to the Heils. This maintains coherency, staging, and imaging. I really don’t fret about a wide dispersion as I always sit in the sweet spot, and a narrower dispersion results in less wall bounce for better staging and imaging.

So lest someone think fewer ways means less cone area, think again. Some of the biggest speakers are 2-ways.

Actually, the Heil is 29.76 square inches of tweeter surface.  I was too late on my edit. 

I am under the impression, (which I may be incorrect) that a three way design is superior to a 2 way design.  
 

Wrong - it’s the quality of the parts and implementation.  Some crossovers are done extremely well they seem seamless.   More crossovers “may” offer more opportunities to have a poor crossover, but it’s not automatic.

As usual, best to demo yourself to hear what resonates with you.