Unique Period Prime/Sequence

Sequence
The sequence of unique period primes begins:
 * $3$, $11$, $37$, $101$, $9091$, $9901$, $333 \, 667$, $909 \, 091$, $99 \, 990 \, 001$, $999 \, 999 \, 000 \, 001$, $9 \, 999 \, 999 \, 900 \, 000 \, 001$, $909 \, 090 \, 909 \, 090 \, 909 \, 091$, $1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$, $11 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$, $900 \, 900 \, 900 \, 900 \, 990 \, 990 \, 990 \, 991$, $909 \, 090 \, 909 \, 090 \, 909 \, 090 \, 909 \, 090 \, 909 \, 091$, $\ldots$