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While taking apart my EV3 brick I accidently pulled on the ribbon cable connecting the screen to the motherboard too hard and it broke. 
I figured I could just get a replacement fpc ribbon cable with the same number of pins and swap out the cable, but turns out the other end is soldered to the screen, so that isn't going to work.
I looked around a lot on the Internet and found several other people with the same issue, but no solutions to the problem. Any advice on what I can do from here?
First of all: before doing anything else, contact LEGO customer support. There's nothing to lose by doing so.
According to the video at https://www.youtube.com/watch?v=gPNJC5Uz9HY , it's possible to replace the screen with a 1.8 inch colour TFT display with the aid of a flat ribbon adaptor.
I'm unsure if you would need to replace the Lego firmware with https://www.ev3dev.org/ or not.
Edit: Lego offers electronic schematics of the EV3 as part of the "Hardware developer kit" that can be downloaded from https://education.lego.com/en-us/product-resources/mindstorms-ev3/downloads/developer-kits . It tells us that the display connects through SPI, and contains this diagram of special relevance:
You should be able to connect any SPI display, which would need 5 connections: +3.3V (or VDDA), 0V (or GND), clock (CLK), MOSI (or DIN or RX in a SPI peripheral), and CS (or SS). Be aware that the nomenclature for SPI is confusing. The screen does not send data back to the board, so a MISO/DOUT/TX connection is not needed.
The stock firmware should work with any monochrome SPI display, but we hobbyists cannot offer you any guarantee at all.
Anyway, the original screen seems to use a pretty standard 18-pin 1.0mm-pitch ribbon cable. You should be able to unsolder it and replace with a new one. But since the connector at the board is a 18-pin 0.5mm-pitch, you'd need to use an adaptor like the aforementioned one.
Bear in mind that soldering tracks with a 1.0mm pitch requires precision soldering skills (and you can imagine that 0.5mm pitch will just be harder). I strongly recommend practicing your soldering skills first.
Edit: According to the folks over at the Electronics stackexchange, it's possible to repair a flat ribbon cable.
