If you have males and females of the blues and the greys you can get there either way. I don’t know that one way is that much better than the other. In either case the offspring will be split for the blue egg gene. The males will be split for barring. So you can take you best male of the offspring and mate if back to the hens. If you mate him to a blue hen, in the F2 generation half the offspring will be barred, male and female. Half the pullets will be pure for the blue egg gene and half will be split for the blue egg gene.
If you mate that F1 cockerel to a grey, half the pullets will be barred, half the cockerels will be pure for barring and half the cockerels will be split for barring. Half the pullets will lay white eggs and half will lay blue, but they will be split for the blue egg gene.
From the F2 generation and forever afterwards, only hatch blue eggs. Without doing test hatching you can’t tell if a hen is pure for blue or is split since it is a dominant gene. But by only hatching blue eggs, you strengthen that blue egg gene.
Always keep only barred pullets F2 and beyond. The barred gene is also dominant, but two copies of barring normally gives you a lighter cockerel than one with split barring. So keep your lighter colored cockerels to breed. You should be able to get to pure barred birds pretty quickly. The blue egg gene is harder to get pure.