I play the white side of this position often, and have an incredible winrate because to be honest, it feels very hard for black to develop naturally. Here's an example of how things can go wrong:
1. e4 c6 2. d4 d5 3. e5 Bf5 4. h4 h5 5. Bd3 Bxd3 6. Qxd3 e6 7. Bg5 Qb6 8. Nd2 c5 9. c4 Qxb2 10. Rd1 Qxd4 11. Qb3 Qxe5+ 12. Ne2 Qc7 13. cxd5 exd5 14. O-O
Black is able to win several pawns, but in order to "punish" white's play has to move their queen many moves in a row. While it appears black has done nothing "wrong," this position just so happens that white gets a huge initiative and is practically winning already because of the open lines against black's king and huge lead in development. Stockfish only thinks this is +0.6 or so, but as you keep playing through the variations, you'll see the evaluation rise and rise, sometimes even with best play from black. Leela loves this position for white. Here is an example game I played in this line that does not end well for the black player.
A more practical alternative for black is to try and trade the queens off instead of pawn grabbing. This is typically done with the maneuver Qa5+->Qa6 (or Qb6->Qa6) so that Qxa6 can be met with Nxa6, supporting a c5 break. Here's the beginning of a game between Nepo and Motylev, 2014 (https://lichess.org/0y7cwwGs) which black won:
1. e4 c6 2. d4 d5 3. e5 Bf5 4. h4 h5 5. Bd3 Bxd3 6. Qxd3 e6 7. Bg5 Qb6 8. Nd2 Qa6 9. c4 Bb4 10. b3 c5 11. Nf3 Nc6 12. dxc5 Bxc5 13. O-O Nge7 14. a3 Ng6 15. b4 Ncxe5 16. Nxe5 Nxe5 17. Qc3 Bd6
You can see how pinning the c4 pawn to the d3 queen means that black's center is more secure: the c5 break has the added benefit of not allowing for a later cxd5 exd5, opening up black's central king. White still had some play in the final position, but collapsed after a premature f4, which ultimately opened up the white king rather than opening up the black king. Black's play revolves around the pressure on white's center, which can sometimes collapse (and sometimes should collapse in order to expose black's king) and lead to double-edged positions that black has winning chances in.
To be honest, I think black's best line here is:
1. e4 c6 2. d4 d5 3. e5 Bf5 4. h4 h5 5. Bd3 Bxd3 6. Qxd3 Qa5+ 7. Nd2 (7...Qa6?! 8. e6!) e6 8. Nf3 Qa6
so that we get the same sort of play as in the previous game but ensure that white's bishop is stuck still on c1. Vidit won a brilliant game against Andriasian as black in this position in 2018 (https://lichess.org/QAd7qBrk).
Finally, to be complete, but at the risk of giving away some cutting edge opening theory, I think white can improve upon this opening by following a Leela recommendation:
1. e4 c6 2. d4 d5 3. e5 Bf5 4. h4 h5 5. Bd3 Bxd3 6. Qxd3 Qa5+ 7. b4!? (7...Qxb4+ 8. Nd2 e6 9. Rb1 Qe7 10. Ne2) Qa6 8. e6! Qxd3 9. cxd3 fxe6
There's a whole world of unexplored territory in this line, but play in it is unlike anything else I've ever seen, so I don't have any amazing recommendations for black except that taking the b pawn seems risky. Currently, this b4 pawn sacrifice seems to have been played only twice in 2018. Time will tell if it catches on in the future! I had a fascinating game from this position recently, where I won the opening battle but ultimately lost on time: https://lichess.org/gUqyAa1u/white. EDIT: The cat is out of the bag because on November 4, 2021, Jorden Van Foreest unleashed this line against his unsuspecting 2600 GM opponent at the FIDE Grand Swiss 2021 event. Check out how he won in <15 moves here.
Enjoy, and good luck in the Caro!