How to capture significant increases in one category("class") but not the rest? 
Let's say I have like 300 probes. For Probe 1, you can see the treatment effect on Class 2 but not the others. In probe2, we can see treatment effects in both Class2 and Class3 which is no good. Out of the 300 probes, I would like to find a pattern (specificity) like probe 1 (effects on one class but not the other).
What kind of method/metrics/models can I use? I'm new to statistics/DS and my main language is R so if you can direct me to the library or function that'll be great too.

IIUC, you already know the treatment effects (i.e. you already have the data for those plots you have posted) but you have 300 probes and you want an automatic way of finding those where the largest effect is also more or less the only larger effect.
In this case, I would suggest computing for each probe the standard score of the largest effect $e_{max}$: compute the mean $\mu_e$ and standard deviation $S_e$ of the effects and then compute:
$$
z_{max} = \frac{e_{max}-\mu_e}{S_e},
$$
which is the distance to the mean in units of the standard deviation and gives you a comparable measure of how much $e_{max}$ "sticks out".
This leaves you with as many standard scores $z_{max}$ as there are probes and then you can just pick those with the largest ones.
Of course, you could also try other scores, e.g., denoting the second largest effect as $e_{sl}$ you could try as score:
$$
s = \frac{e_{max}-e_{sl}}{e_{max} - \mu_e},
$$
which would make sure the relative distance between the largest and second-largest is especially large.
