For me, a variety $X$ is assumed to be irreducible and normal over some algebraically closed field $k$ of characteristic zero.
I call $X$ a toric variety if it has an algebraic torus $T$ which embeds $T\hookrightarrow X$ as an open subset such that the action of $T$ on itself (multiplication) extends to an algebraic action on all of $X$.
I call $X$ a flag variety if $X\cong G/P$ for some connected reductive linear algebraic group $G$ and some parabolic subgroup $P\subseteq G$. Note that flag varieties are smooth and projective.
My question is this:
Which varieties $X$ are simultaneously a toric variety and a flag variety?
Certainly such varieties $X$ need to be smooth and projective. Some examples:
(1) If $X=\mathbb{P}^n$, then this is certainly a toric variety, and it is a flag variety via $SL_n/P_n$ where $P_n$ is a maximal parabolic subgroup (not equal to all of $SL_n$).
(2) Products of the above example, e.g. $\mathbb{P}^1\times\mathbb{P}^1$, since toric and flag varieties are closed under taking products.
(3) Points since a point is clearly toric (the trivial torus action) and is of the form $G/G$.
(4) As a non-example, the homogeneous space $SL_3/B$ is a flag variety but not a toric variety ($B$ is the Borel subgroup). Indeed, it is clearly a flag variety, but its Cox ring is not a polynomial ring, so it cannot be toric (see Corollary 2.10 in "Mori dream spaces and GIT" by Hu and Keel; and see Example 4.1 in "The Cox ring of a spherical embedding" by Gagliardi for the Cox ring calculation).
I'm not sure if there is a "nice" general answer to my question. Any help is appreciated.
Partial flag varieties satisfy a property called "being $\mathscr D$-affine". (This is the Beilinson-Bernstein localisation theorem). The following article "$\mathscr D$-affinity and toric varieties" prove that the only projective toric varieties that are $\mathscr D$-affine are product of projective spaces.