Given I have the most primitive spherical harmonic, $Y_{00}(\theta, \phi)=\frac{1}{2} \frac{1}{\sqrt{\pi}}$ and I look at one of the three second most primitive ones, e.g. $Y_{11}(\theta, \phi)=-\frac{1}{2} \sqrt{\frac{3}{2 \pi}} \sin \theta e^{i \phi}$, it seems to me that when I look at their plot that I did with Mathematica,
that I would be able to approximate $Y_{11}$ with $Y_{00}$ by putting two of them next to each other.
How can I find an (analytical?) solution that lets me express the higher order spherical harmonic in terms of two lower order spherical harmonics?
Slightly too long for an answer: it definitely depends on the norm you are using for your functions. If you are using the $L^2$ norm induced by round metric on the sphere, I have very bad news: the approximation is going to be very bad in general. This is because spherical harmonics are orthogonal with respect to this norm. Let us see a baby example, for instance approximate $Y_{10}$ by $Y_{00}$, and $$\|Y_{10}-\lambda Y_{00}\|^2= \|Y_{10}\|^2+\|Y_{00}\|\lambda^2 + \lambda \langle Y_{10},Y_{00}\rangle = \|Y_{10}\|^2+\|Y_{00}\|\lambda^2$$ Minimizing with respect to $\lambda$ gives that the best approximation is... $0\times Y_{00}$.
For your general case of approximating $Y_{11}$ it will be the same, because the set $\{Y_{lm}\}$ is orthonormal!