A playful blend of 'bear' and 'béarnaise' (a classic French sauce), likely a humorous or invented culinary term with no standard definition.
A portmanteau or pun combining 'bear' with 'béarnaise' sauce, which is named after Béarn, a region in France; this appears to be a modern coinage, possibly used humorously in food writing.
Culinary puns and portmanteaus like 'bearnaise' show how creative English speakers get with language when talking about food—mixing animal names with classical sauce names creates instant, amusing new dishes that don't actually exist but perfectly convey the joke.
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