Answer: C

Small Hint:

Use $TTT=111T=3\cdot37\cdot T$

Big Hint:

The prime $37$ must divide one of the two-digit factors; test its two-digit multiples ending in the common digit $E$

Solution:

Since $$TTT=111T=3\cdot37\cdot T,$$ the prime $37$ divides one of $YE$ and $ME.$ A two-digit multiple of $37$ is $37$ or $74.$ The value $74$ is impossible: the other two-digit factor ending in $4$ is at least $14,$ and $74\cdot14\gt999.$ Hence one factor is $37,$ so $E=7.$

The units digit of the product is the units digit of $7^2,$ so $T=9.$ Thus the product is $999,$ and the other factor is $$\frac{999}{37}=27.$$ The four digits are $2,$ $3,$ $7,$ $9,$ whose sum is $21.$

Therefore, the correct answer is C.