What do you mean "is drawn from"? Isn't the previous condition a reasonable characterization of "you know what distribution it's drawn from"?

Let's say you draw 1 million samples from a uniform distribution between 0 and 1. I'm saying that if you know the distribution you can sort in O(N log(log(N))). Are you claiming you can sort them in O(N)?

Assuming "1 million" is a proxy for taking the limit to infinity, i.e. the usual asymptotic setting, if you are willing to accept that the algorithm will "only" work with probability 1, then yes, I can.

Let N be the size of the array. Create N new empty lists, numbered from 0 to N-1. Divide [0, 1) in N equal brackets B0 = [0, 1/n); B1 = [1/n, 2/n]... BN-1 = [(n-1)/n, 1). For each element, determine the bracket it falls in and append it to the corresponding list. Sort each list. Because of the uniformity hypothesis, for n -> infinity, the probability that each bracket contains O(1) elements goes to 1, therefore the running time is O(N) with probability 1.