The answer appears to be yes, that we can construct such numbers at present. The techniques that have been used recently have their roots around 1985 when elliptic curves were first applied to cryptography and factorization and when personal computers with RAM by the megabyte became common.
I would like to thank Charles for reminding me that a product of exactly two primes is called a semiprime.
Chris K. Caldwell, a professor at the University of Tennessee at Martin whose current research interest is prime number theory, writes that "small examples of proven, unfactored, semiprimes can be easily constructed." What is easy for him is not so easy for me, but it might not be too hard if I would re-read my copy of Bressoud's Factorization and Primality Testing.
Proven, unfactored semiprimes are called "interesting semiprimes" by Don Reble, a software consultant who took up the problem from (at least his interpretation of) remarks by Ed Pegg, Jr. There are at least two examples online, a 1084-digit interesting semiprime constructed by Don Reble and a 5061-digit interesting semiprime constructed by David Broadhurst, a theoretical high energy physicist.
Reble's interesting semiprime is in a text file that presents some parameters for a proof and the proof itself. It relies on properties of elliptic curves and is therefore currently over my head. Part of Reble's proof is that his semiprime survives a check that it is not a base-two strong probable prime.
Broadhurst's interesting semiprime is in a text file that can be input to Pari. He has written there the relatively elementary conditions and the parameters that he used in order to prove that his number is a semiprime, basing his work on Reble's. He provides the location of a certificate that one of his parameters was proven prime using the free-of-cost, closed-source program Primo by Marcel Martin. Primo is an implementation of elliptic curve primality proving. For suggesting the problem, Broadhurst thanked Reble and Phil Carmody, a Linux kernel developer and researcher in high-performance numerical computing.