I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In other words, consider $\Lambda \in \mathcal{P}(\mathcal{P}(X))$. Then, an empirical measure for $\Lambda$ has the form: $\frac{1}{N}\sum_{n=1}^{N} \delta_{\mu_i}$, where $\mu_i$ are i.i.d. random variables taking values in the space $\mathcal{P}(X)$ distributed according to $\Lambda$.

Now, given a specific outcome $\mu_i (\omega)$, we can again draw samples from $\mu_i (\omega)$ by considering some i.i.d. random variables $X_i$ distributed according to $\mu_i(\omega)$, and then form the empirical measure $$\hat{\mu}_i:=\frac{1}{M}\sum_{n=1}^{M} \delta_{X_i}$$. Then, it makes sense to think of the measure $$\frac{1}{N}\sum_{n=1}^{N} \delta_{\hat{\mu}_i}$$ as a "doubly empirical" measure for $\Lambda$, in the sense that we have drawn samples from $\Lambda$ and then drawn samples from those samples.

My question is: where in the literature has this object been considered? Does it already have a standard name in the literature? Googling has turned up nothing for me, nor does digging around in Kallenberg's Random Measures book.