Guys, I have done carpentry work and cabinet work. The hardware, like knobs and handles is often supplied with long screws because the doors can easily be 3/4" thick or more. But then, some may be thinner. And if they are used on a sheet metal door, then the thickness may be less than 1/32". And carpenters or cabinet makers do not always have the means to cut a long screw down. AND, supplying two or three screws is an added expense which would make the product more expensive.
So a long screw with a deep hole is one answer to making it fit on whatever door or other thing it may be installed on.
What I find strange here is that no one is asking what STYLE of tap he is using: straight flute, spiral flute, or spiral point. He has a BLIND hole so he should be using spiral flute taps. At least that would be my first thought. I suspect he is using a straight flute because he says it is a bottoming tap. And a straight flute may be pushing some of the chips ahead of the tap. Or it could be a spiral point tap which is designed to push the chips ahead of the tap. Either could/would prevent it from going the full 3/4" into the hole. And if he tries to run it that deep, then it stands a good chance of breaking.
To the OP:
I would suggest a spiral flute tap as a first try.
McMaster-Carr is the complete source for your plant with over 595,000 products. 98% of products ordered ship from stock and deliver same or next day.
www.mcmaster.com
You can start with a bottoming chamfer spiral flute and see if that works. Cabinet hardware is often made of softer metals and it may be OK.
But if you still have tap breakage you may need to start the hole with a plug chamfer spiral flute tap and then blow it out and finish with the bottoming chamfer one. This can be more efficiently done if a batch is first tapped with the plug chamfer and then switch to the bottoming chamfer.
Another point is that broken taps are often, dare I say most frequently, due to a misalignment when the tap is started. You do not say how the hardware is held in the drill press or if you drill a batch first then tap or drill and then tap on a one by one basis, but it is critical to ensure that the tap is aligned both in the X-Y directions and in the angular sense with the drilled hole. If you try to force it in off center or at an angle it may work for the first turns but the error will increase as you approach the bottom and breakage will become more likely. And at the end is where you say you are breaking them.
I provided a link to McMaster's page on spiral flute taps as a reference, but there are a number of good brands available. Whatever you do, DO buy a quality brand of GROUND thread tap.
A thread forming tap may also be a good choice. I have little experience with them so I will leave that part of the discussion to others.
HSS vs. cobalt? You can look it up but basically the addition of cobalt to HSS makes it harder, tougher, and stronger. And it can work at higher temperatures. Anyway, that is the general idea. Which is better for this may need to be answered by cut and try.