How to add an annotation that calculates a value as part of plot composition?

Using ggplot, I want to add an annotation that describes the mean value of an axis, and I wish to make this calculation on the fly, rather than a preemptive calculation that assigns a value to a dedicated object.

Example

Consider the following carat histogram by cut:

library(ggplot2)
library(dplyr)
#> 
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#> 
#>     filter, lag
#> The following objects are masked from 'package:base':
#> 
#>     intersect, setdiff, setequal, union

diamonds |> 
  filter(depth > 60) |> 
  ggplot(aes(carat)) +
  geom_histogram() +
  stat_summary(aes(xintercept = after_stat(x), y = 0), 
               fun = mean, 
               geom = "vline", 
               orientation = "y", 
               color = "red") +
  facet_wrap(~cut)
#> `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

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^{Created on 2023-07-13 with reprex v2.0.2}

Desired Output

Can I tuck another line of code to the ggplot composition that will add an annotation for "Mean Carat = mean(carat)" and it will do the mean calculation per facet?

Pseudo Plot

It's basically the same as for the vline but instead we have to use geom="text" and map on the label aes:

library(ggplot2)
library(dplyr, warn = FALSE)

diamonds |>
  filter(depth > 60) |>
  ggplot(aes(carat)) +
  geom_histogram() +
  stat_summary(aes(xintercept = after_stat(x), y = 0),
    fun = mean,
    geom = "vline",
    orientation = "y",
    color = "red"
  ) +
  stat_summary(
    aes(
      label = after_stat(paste0("Mean Carat = ", round(x, 3))),
      y = 6000
    ),
    fun = mean,
    geom = "text",
    orientation = "y",
    color = "red",
    hjust = 0,
    position = position_nudge(x = .1)
  ) +
  facet_wrap(~cut)