Dplyr mutate minimum column name

Question

I have a df with many columns but for this issue am focusing on four date columns A, B, C, D. I am using the following to create a column for the minimum date for that row.

mutate(MIN_DATE = pmin(A,B,C,D))

To create a column for the minimum date for that row.

  A         B       C       D       MIN_DATE
1-1-20   1-2-20  1-10-20  2-10-20    1-1-20
1-2-20   1-4-20  1-11-20  2-1-20     1-2-20
1-3-20   1-5-20  1-20-20  2-3-20     1-3-20

What I would like to do is add another column which is the column name of the minimum date which would look like :

  A         B       C       D       MIN_DATE   MIN_COL
1-1-20   1-2-20  1-10-20  2-10-20    1-1-20       A
1-2-20   1-4-20  1-11-20  2-1-20     1-2-20       A
1-3-20   1-5-20  1-2-20   2-3-20     1-3-20       C

What is the easiest way to accomplish this, some combination of names and which.min()?