It is distinctive of the Mormon religion that among modern people in modern societies they have above-replacement fertility and larger families among the more educated and more prosperous.
How is this done?
My feeling is that it is an outcome of the religious emphasis on the family.
For Mormons, the married couple (whose marriage has been sealed in a Temple) is the unit of the highest form of eternal salvation, and their family remain with them in the afterlife and through eternity.
This is, I think, an unique emphasis among world religions. And from it many things follow:
Marriage and family become the focus of life - early marriage is encouraged (and institutional arrangements are made to enable this).
The basic social arrangements (including the family) are patriarchal; which history tells us is stable and viable.
Families will want to have as many children as they can afford to raise decently since they are future companions in the afterlife - and spiritual guidance concerning this will be available to them through prayer and fasting.
Since the family is the primary social unit, the family needs to be economically autonomous (as a general rule or aspiration) - so families need to live within their means, without borrowing or doles (since sustained cross-subsidy would damage other families).
And since Mormons control fertility by the use of contraception, consequently the largest families are among the wealthiest.
In sum, most of the distinctive features of Mormon fertility seem to derive from the distinctive theological emphasis on the family.
That this distinctive fertility pattern is indeed an aspect of the Mormon church seems to be confirmed by the fact that fit seems to be the same both in the US and the UK.