There is some confusion in the way the question is set up. You have to begin with a fixed Borel subgroup $B$ in order to speak about a "simple" reflection in the Weyl group. Then there is a unique "minimal" parabolic subgroup $P \supset B$ corresponding to the specified simple root/reflection. This in turn has a Levi subgroup of rank 1, while $P/B$ is naturally isomorphic to the projective line. Of course, $P$ itself contains many other Borel subgroups besides $B$, but now the problem has shifted. All such Borels contain the radical of $P$. so after factoring that out you are just looking for at the flag variety of a rank 1 group which is the desired copy of the projective line.
There is some confusion in the way the question is set up. You have to begin with a fixed Borel subgroup $B$ in order to speak about a "simple" reflection in the Weyl group. Then there is a unique "minimal" parabolic subgroup $P \supset B$ corresponding to the specified simple root/reflection. This in turn has a Levi subgroup of rank 1, while $P/B$ is naturally isomorphic to the projective line. Of course, $P$ itself contains many other Borel subgroups besides $B$, but now the problem has shifted. All such Borels contain the radical of $P$. so after factoring that out you are just looking for the flag variety of a rank 1 group which is the desired copy of the projective line.