Can you react to the Gutter Palace "at the start of the beginning phase" triggered ability?

Yes, you can react to the triggered ability of Gutter Palace.

According to Rule 383.3, when a triggered ability's condition is met, it behaves like an activated ability and is placed on the chain. Because the ability is placed on the chain, it creates a Closed State, which allows players to play cards or activate abilities with the [Reaction] keyword in response.

Sequence of Events:

1. Trigger: At the start of your Beginning Phase, the condition for Gutter Palace's ability is checked. If the conditions are met, the ability triggers and is added to the chain.
2. Priority: Once the ability is on the chain, priority passes to the players.
3. Reaction: You may play a [Reaction] card or activate a [Reaction] ability to change the game state.
4. Resolution: The chain resolves in reverse order (LIFO - Last In, First Out). If your reaction successfully changes the game state so that the conditions for Gutter Palace are no longer met, the ability will resolve but fail to trigger the win condition.

This interpretation is consistent with Rule 383.3 and the general mechanics of triggered abilities in Riftbound. This follows the same logic as other beginning-of-turn triggers, where the game state is checked upon the resolution of the triggered ability, not when it is placed on the chain.

Note: I could not fully verify this ruling against the retrieved rules.