I don't believe it is possible to achieve this using only hoppers, at least not without a crap-ton of complicated redstone. However, if you're willing to add in a simple rail system and a chest minecart, you can do it quite easily. Here's the setup:
![enter image description here](https://i.stack.imgur.com/Hn4Xj.png)
Place rails on top of your hoppers, and on one end (left in the image) place a powered rail and a redstone torch to power it continuously. On the other side (right in the image) place a detector rail connected to a redstone comparator, which feeds a signal to a redstone repeater and then another powered rail (under the minecart in the image). Place a chest minecart on the powered rail and a button on the block underneath.
Then, when you have a whole lot of a certain block/ore to smelt, place it in the chest minecart and just hit the button. The minecart will bounce back and forth on the track, distributing its load evenly amongst the furnaces and eventually coming to rest back at the start when it is empty. This works because the detector rail and comparator will create a redstone signal whenever a non-empty cart passes over the rail, activating the right-side powered rail and sending the minecart back out.
BONUS: You can further improve this system by placing a hopper chain below your furnaces that automatically pulls out all the smelted results and places them in a chest. It's also possible to add a second track system to feed combustibles into the backs of your furnaces in an evenly distributed manner as well. This screen shot shows one such system I added to the above example:
![enter image description here](https://i.stack.imgur.com/mAKbJ.png)
Note that I had to add an extra row of hoppers to the back side of the furnaces in order to create a separation between the tracks (or else they tended to try and connect to one another).
Well, i tried your system in my redstone world, and i managed to reproduce the problem.
In fact, some droppers randomly do not shoot. Most of the times the same three, but not always. I thought it was some kind of issue related to some block upload detection...
But not worry my friend, i managed to fixed it! Just a line of repeaters and it's done.Not so elegant, but it works fine. See this.
![enter image description here](https://i.stack.imgur.com/XObe4.jpg)
Now it shoots exactly 16 items each time the button is pressed :)
May i take your time for a longer answer?
First, i couldn't reproduce your problem, as stated before by @QbsidianH20
Second, after your first filling of the droppers, seeing your hoppers configuration, you will gradually start to make useless the top ones, then gradually the bottom ones too, starting from the far right (facing the front of the droppers), eventually ending with only the first one or two shooting. Tis is because of the way hoppers work, prioritizing vertical transfer over horizzontal.
This is true for each hopper system with only one input of items, like i see in your configuration.
Moreover, hoppers can only transfer items at a pace of 2.5 per second, making that huge set of droppers completely wasted, sadly.
If you have more items input from different hopper chains, i will suggest you to make a single dropper with clock for each input, maximizing your efficiency both of resources and time.
With your permission, i will proceed to illustrate.
First case: ONE INPUT
![](https://i.stack.imgur.com/o7lmr.jpg)
I copied your configuration. You can see after a while only the first dropper fires, but they are all powered! Fact is, he is the only one with something inside to fire. Poor guy.
Possible solution for one input:
![enter image description here](https://i.stack.imgur.com/Nzont.jpg)
Mono input, mono dropper. The clock on its back is a pretty nice one, only triggers when something's actually inside, staying off when empty. And still quite fast.
If you have a HUGE and i mean reeally huge amount of items to funnel, i suggest to simply copy this configuration as many times as needed.
Like this:
![enter image description here](https://i.stack.imgur.com/JgaUu.jpg)
You can see THREE INPUTS of items into three different droppers, each one with its own clock to minimize unnecessary ticks.
For each dropper we have five hopper lines, all converging into one input.
I have a humongous mob farm with a ridiculous amount of item drop per second, and this is my system. Three droppers don't even need to work all the time.
I hope i managed to help you, good luck building the rest of your farm! ;)
My farm and the funnels for sauce, with the firing mechanism (you can see it shooting items in the drop pool below, in order to be sorted).
![enter image description here](https://i.stack.imgur.com/Kk5TR.jpg)
Best Answer
That aside, with the redstone block on top, the hopper is now ready to receive items. When an item is detected by the comparator, it pushes the redstone block down, which turns the redstone torch on and locks the hopper. To reset it and enable the hopper to again accept one more item, just provide a short pulse to the bottom piston to push the redstone block back up.
This solution is also one wide and tileable, like my previous one. Like my previous one, you will also have to limit the rate the items flow in (maybe by slowing down the dropper clock in the top left of your screenshot?).