You could go the computer engineering route. It's a nice combination of EE and CS classes that will give you electrical theory experience that a CS major will not get and more experience with digital/software design than an EE will get. Not every university offers it. There is usually a large focus on writing C and Verilog/VHDL in the higher up classes. If you can't pursue a computer engineering degree at your school, an EE degree with a heavy focus on digital design will get you close. If can, try to get approval to take CS classes to expose you to more FPGA things if your CS path does that. There are a bunch of free online MIT classes for this kind of stuff. One thing to help, go out and buy a cheap Xilinx or Altera FPGA to start messing around with stuff on your own. Learn how to write fast assembly for MIPS or 8051 and fast C for RISC-V or ARM processors.

MSCoE here. FPGAs have limited applicability due to their high cost per unit and relatively low clock speeds. They were all the rage ~15 years ago during my grad study, but then flattened out. Although, it does look like they're bouncing back, likely due to newfound popularity for mining, and cheaper/better product lines coming out. Cheap, embedded stuff based on MIPS, ARM, even mobile processors like NVidia Tegra, will probably enjoy an order of magnitude broader market footprint, however.

I work in aerospace, FPGAs are the dominant chip. Everything from basic microprocessors to complex SDRs all run on them. Xilinx is the big one but there are smaller shops making more application-specific chips.

The EE degree is only there to make you employable as an EE. You'll have to teach yourself formal methods on your own. That's why it's a way to stand out.

"You have to teach yourself on your own" is always a good idea, but it would be an immense waste of time not to use the classes that your educational institution provides. Double majoring exists specifically for this purpose.