Hence the pitch of the sonic boom should be high.
As others have already pointed out, a sonic boom is not subject to the Doppler effect because it's just a nonlinear pressure pulse (i.e., a single or half period sound wave) due to something moving air faster than the speed of sound.
But most of the times I hear a sonic boom it sounds like a low pitched thunder. Why is that?
A sonic boom, when transformed to frequency space using a Fourier transform, has a very broadband response function (for a pure discontinuity or step function, the Fourier transform would have power at all frequencies). For the sake of simplicity, let's assume there is equal power at all frequencies (this isn't true but it's an easy way to understand the phenomena)
The frequency at the receiver (i.e., pitch heard by an individual) is dependent upon several things, one of which depends upon distance from the source (as previously noted). As you move further from the source, any broadband sound will shift to lower and lower frequency due to the higher frequencies being attenuated.
Another issue is that the human ear does not have a flat frequency response. Very strong sonic booms will sound different than weaker ones due to saturation at some (or all) frequencies within the range of human hearing. So in our overly idealized scenario of a short duration white noise pulse, if the amplitude of the white noise is large enough some ranges of frequency will saturate and cause the brain's interpretation of the "pitch" of the sound to be different than if the amplitude of the white noise was lower.
Note that the frequency response of a sonic boom depends upon the wavelength of the pressure pulse. The sharper the gradients the broader the range of sound frequencies.
Fun Side Note
I have heard sonic booms from whips, bullets passing close by, and fighter jets flying overhead. All of these sound like hollow cracks, not a high or low pitched noise. If you are farther from the source (e.g., the jet breaks the sound barrier at a very high altitude), then it can sound like a lower frequency rumble (the rumble versus crack would require another question and answer).
A few years ago we had an audification specialist working in our lab turning spacecraft data into sound files. At first glance it seemed like a superficial, subjective art project. After chatting with him, I started to realize they weren't superficial or subjective but rather had a very useful analysis technique. So he started listening to magnetic field data in the solar wind without knowing really anything about the data or phenomenon therein. He quickly found some very interesting sounds and we realized a few things. First, the ears can digest and differentiate a much larger bandwidth of information per unit time than the eyes. That is, he was able to search through and consistently find interesting time intervals orders of magnitude faster than anyone in our lab could do "by eye." He was able to categorize and itemize intervals for ~20 years of Wind magnetometer data in a matter of a few weeks. Trying to do the same thing "by eye" would have taken even the fastest of us several years.
Second, he was able to identify the unique sound of collisionless shock waves. They all had a similar auditory response, i.e., a kind of hollow crack and/or thud. When examined in his high end audio software, they all looked exactly as one would expect. They were an isolated band of power across the entire frequency range observed by the instrument.