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Option 1: Nagpur Session of 1959 - This option is incorrect. The Nagpur Session of 1959 did not officially accept the `socialistic pattern of society`.
Option 2: Avadi Session of 1955 - This option is correct. The Congress officially accepted the `socialistic pattern of society` at the Avadi Session of 1955. During this session, the resolution on socialism was adopted, marking a shift in the Congress Party`s vision and ideology.
Option 3: Bombay Session of 1953 - This option is incorrect. The Bombay Session of 1953 did not officially accept the `socialistic pattern of society`.
Option 4: Calcutta Session of 1957 - This option is incorrect. The Calcutta Session of 1957 did not officially accept the `socialistic pattern of society`.
In conclusion, the correct option is 2, as the Avadi Session of 1955 is where the Congress officially accepted the `socialistic pattern of society`.