This is perhaps a naïve question; given an irreducible scheme X, is there a general procedure to find a flat family $Y \to T$ such that over some point $t_0$ we have $Y \times_T {t_0} \cong X$ but for every other point $t \in T$ the scheme $Y \times_T t$ is a disjoint union of reduced schemes?

The example I have in mind is $X = Spec\ k[\epsilon] / \epsilon^2$, $Y = Spec\ k[x, y] / (y^2 = x) \to T = Spec\ k[x]$.

This is perhaps a naïve question; given an irreducible scheme $X$, is there a general procedure to find a flat family $Y \to T$ such that over some point $t_0$ we have $Y \times_T {t_0} \cong X$ but for every other point $t \in T$ the scheme $Y \times_T t$ is a disjoint union of reduced schemes?

The example I have in mind is $X = \textrm{Spec}\ k[\epsilon] / \epsilon^2$, $Y = \textrm{Spec} \ k[x, y] / (y^2 = x) \to T = \textrm{Spec}\ k[x]$.