Unfortunately no. When I say custom designs I mean custom circuits using a MCU and other ICs to do a specific task.
It is my understanding that you first design your circuits using off-the-shelf MCUs and ICs to do your task. When you need higher performance you move the design over to a the FPGA solution. Then finally when you need even more performance you switch over to an ASIC. An ASIC is essentially designing your own processor. They are custom ICs, manufactured on wafers, to do your task.
If you want to get into designing custom processors you should look into FPGAs, VHDL, and Verilog. This will be your cheapest option to start learning how to design custom processors. ASICs have huge upfront NRE fees starting around $500,000.
For cheap FPGA started boards check out the Mojo FPGA Development Board and the DE0-Nano - Altera Cyclone IV FPGA starter board (www.adafruit.com).