Ezreal Build

finish manamune first for faster stacking and stacks on auto. tear + triforce is weak because it sets your triforce back a full 720 gold which makes the trinity spike a lot weaker than it should be, so you may as well rush the muramana since that's where your true power spike lies.

Tear to Tri to Manamune

Tear, manamune then triforce. Then bloodthirster, finish cdr boots, last whisper, then finish with a "bork, GA, Bansheees, Zhonyas"

It is actually really a tossup. It tends to go like this:

Early First back: Tear + whatever else you can afford.

Second back: If you are ahead and have the gold, finish Manamune. If you aren't ahead or need a power spike, Sheen if Dragon fight/other poke fight, Phage if need to survive all-ins.

Third back: Trinity(Or components if you can't afford) if you went Manamune at second back. If you went Trinity component and are now winning, Manamune. If you are still behind, finish Trinity/more components.

If you go Tear->Trinity->Manamune you can get Muramana to pop 23-25 minutes in, faster if you are fed blues because you are mid Ezreal.
If you go Tear->Manamune->Trinity you are a bit weaker in the buildup, but you can hit that Muramana spike at 20-23 minutes.

So, it comes down to a number of factors, including if you are capable of abusing a 3-5 minute earlier power spike at the cost of some early game power? Are you winning already? Do you need HP to survive the LeBlanc allinning you constantly? Are you getting into early dragon fights or tower sieges where you really want that Sheen damage? I personally like the Muramana first.

And this all isn't even taking into account the OTHER big early game choice. That one where you realize the enemy has a lot of AD threat or a lack of gap closers so you build . Or the third/fourth item debate of BotRK vs Bloodthirster.

(Edited in PS: My favorite build is with Debonair skin. Color code that shit.)