In my recent post on Romans 2, I speculated whether Paul takes the phrase 'doing the law' as shorthand for living according to the obedience of the covenant in a general sense, whether he is speaking of the Mosaic or the New. In the Old Covenant, this would not only include ethical obedience, but performing the sacrifices of the Law.
In 1 Peter, which is thoroughly Hebraic (even if you take diaspora as metaphorical in 1:1), we find:
Peter, an apostle of Jesus Christ to the chosen refugees of the diaspora...according to the foreknowledge of God the Father, in the holiness of the Spirit, in obedience and the sprinkling of the blood of Jesus Christ; may grace and peace be multiplied to you. 1Peter 1:1-2 (my translation)
The terse style of 1 Peter (verbs? what verbs?!) leads to plenty of interpolation in the standard translations, e.g. that you may obey Jesus Christ and be sprinkled with his blood, NASB(!). Anyway, Peter's thought here is of his audience living as Christian refugees (or aliens) by God's election (thoroughly Jewish), in the holiness of the Spirit (parallel to Pauline definitive sanctification), in obedience and the sprinkling of blood (the New Covenant life). It strikes me that the thought here is compatible with (and derived from) the Old Testament covenant life of 'doing the Law'. The fact that Peter has obedience prior to sprinkling of blood poses a problem for those who want (so desperately!) to see the ordo salutis reflected here, but I think that Peter's thought is that these Jewish Christians have been called to New Covenant obedience in the Spirit and that their sins (which they still commit) are atoned for by the blood, not the blood of animal sacrifices that cannot remove sin, but the blood of Jesus the Messiah, the blood of the New Covenant (the argument of Hebrews). So, we can see parallels between Old and New Covenant obedience (what Paul calls 'doing the law'), alongside the excellency and significance of the atoning work of Christ.