Short answer: the data can't isolate a sideboard card that clearly helps, and it has no usable sample on Chalice of the void. The best signals are Cavern of souls and, more weakly, Disruptor flute.
I treated the Modern archetype "broodscale" as your deck, since there is no separate "mono green broodscale" entry. It is the Urza's Saga, Eldrazi Temple and Ancient Stirrings list. I compared it against "azorius control" using 2026-01-01 to 2026-10-06 data.
Matchup: Broodscale is roughly even at 105-101 with 17 draws, a 51.0% winrate with draws excluded (223 matches). Azorius control mostly runs Orim's chant main, in 767 of about 925 decks, usually as a 4-of. Silence is rare (17 decks main). Only 18 matches came against builds with 6 or more chant effects, and Broodscale went 5-11 (31%) in them. That is too small to be conclusive, but it points the same way as your concern.
Sideboard splits (Broodscale decks carrying the card, versus Azorius control):
Chalice of the void: 17 Broodscale decks run it in the sideboard, and they have only 3 matches against Azorius control (1-2). I can't say whether it works. Azorius control also doesn't rely on cheap cards that Chalice would hit. Its sideboard is Consign to memory (99% of decks), Mystical dispute (97%) and High noon. These are also unproven against Broodscale.
I don't have data on which cards players actually board in, so treat these as deck-level correlations. They are not strategy advice.
Sources: 444 Broodscale tournaments in the window, 71.8% MTGO and 28.2% Melee.
https://www.mtgo.com/decklist/modern-rc-super-qualifier-2026-10-0312855444
https://www.mtgo.com/decklist/modern-challenge-16-2026-10-0312855475