In a sauce pan, heat the oil and 1 tablespoon of butter. Add shallots, half the tarragon, half the parsley, 1/4 cup of the mangoes and the vinegar. Cook over gentle heat for 20 minutes or until only 1 tablespoon of liquid remains. Strain through a fine mesh sieve.
To clarify butter, melt the remaining 11 tablespoons of butter in a double boiler over hot heat. After several minutes, milk solids will settle at the bottom. Pour the clear butter into a bowl, leaving the milky residue behind.
To finish the Bearnaise, combine the egg yolks and the vinegar mixture in a double boiler over hot water. Whisk well. Gradually incorporate cold water, salt and pepper, continuing to whisk vigorously until the mixture becomes creamy. Remove the double boiler from the heat and continue whisking while adding the clarified butter in a thin stream.
Stir the remaining tarragon, parsley and mango into the mixture. Use Bearnaise immediately or reserve in a double boiler over hot but not boiling water.