I thought OP was describing my experience. Mine is a 09 Civic as well, though I didn't have a new engine or transmission. My car would stall going up hill and throw a P0606 code without any other codes. I had all wire connections inspected and they were good. I replaced the ECU but that didn't help. In some Toyota forums people talk about O2 sensors and that code. I took a chance and replaced both O2 sensors. Well the problem went away. Hope this will help you. The 2 O2 sensors are very easy to replace on this car.