First of all, there's really no way to start capturing events from the database starting at <date>. The connector's can either begin capturing CDC events by starting with a consistent snapshot of the database (which could contain state modified before or after <date>), or by simply starting at the current timestamp.
On the other hand, if you're talking about consumers reading CDC events that represent changes newer than <date>, there are a couple of potential options as long as <date> is after the time at which the connector completed its snapshot.
If the connector had been grabbing individual events prior to <date>, then it might be possible with Kafka 0.10 or later to use Kafka's new feature that records for each message the timestamps it was created (captured in Debezium's case) or logged. In this case, the Kafka 0.10 consumer can say they want to start consuming events given <date>, and the broker efficiently finds that starting point. Note that this is obviously not the same thing as the timestamp used by the database to perform operations, so some filtering would have to be done by the consumer until it saw the timestamp in the CDC event message.
Alternatively, if the connector had been grabbing individual events prior to <date>, then the consumer could just start at the beginning and ignore all CDC events that were performed by the database prior to <date>. This is easier, but could be potentially time consuming to just find the first CDC event at or later than <date>. This would work with all versions of Kafka and Kafka consumer clients.
I hope this helps.
Randall