State.Restarting is not set to true after a `docker restart` command

Description

When a container is to be restarted according to restart policy, State.Restarting is set to true
But when a container is restarted by docker restart command it is not. There's no way by the inspect API to know container is going to restart

Reproduce

$ docker run --name test -d alpine ping localhost
34ce1163ab9a3c4ac920e4f60735977114effbaefd09424d645e965b8da2b1da
$ docker restart test &
[1] 55396
$ docker inspect test -f '{{.State.Restarting}}'
false

Expected behavior

$ docker inspect test -f '{{.State.Restarting}}'
true

docker version

Client:
 Cloud integration: v1.0.31
 Version:           23.0.5
 API version:       1.42
 Go version:        go1.19.8
 Git commit:        bc4487a
 Built:             Wed Apr 26 16:12:52 2023
 OS/Arch:           darwin/arm64
 Context:           desktop-linux

Server: Docker Desktop 4.19.0 (106363)
 Engine:
  Version:          23.0.5
  API version:      1.42 (minimum version 1.12)
  Go version:       go1.19.8
  Git commit:       94d3ad6
  Built:            Wed Apr 26 16:17:14 2023
  OS/Arch:          linux/arm64
  Experimental:     false
 containerd:
  Version:          1.6.20
  GitCommit:        2806fc1057397dbaeefbea0e4e17bddfbd388f38
 runc:
  Version:          1.1.5
  GitCommit:        v1.1.5-0-gf19387a
 docker-init:
  Version:          0.19.0
  GitCommit:        de40ad0

docker info

Client:
 Context:    desktop-linux
 Debug Mode: false
 Plugins:
  buildx: Docker Buildx (Docker Inc.)
    Version:  v0.10.4
    Path:     /Users/nicolas/.docker/cli-plugins/docker-buildx
  compose: Docker Compose (Docker Inc.)
    Version:  v2.17.3-53-g77dc9b54f.m
    Path:     /Users/nicolas/.docker/cli-plugins/docker-compose
  dev: Docker Dev Environments (Docker Inc.)
    Version:  v0.1.0
    Path:     /Users/nicolas/.docker/cli-plugins/docker-dev
  extension: Manages Docker extensions (Docker Inc.)
    Version:  v0.2.19
    Path:     /Users/nicolas/.docker/cli-plugins/docker-extension
  init: Creates Docker-related starter files for your project (Docker Inc.)
    Version:  v0.1.0-beta.4
    Path:     /Users/nicolas/.docker/cli-plugins/docker-init
  sbom: View the packaged-based Software Bill Of Materials (SBOM) for an image (Anchore Inc.)
    Version:  0.6.0
    Path:     /Users/nicolas/.docker/cli-plugins/docker-sbom
  scan: Docker Scan (Docker Inc.)
    Version:  v0.26.0
    Path:     /Users/nicolas/.docker/cli-plugins/docker-scan
  scout: Command line tool for Docker Scout (Docker Inc.)
    Version:  v0.10.0
    Path:     /Users/nicolas/.docker/cli-plugins/docker-scout

Server:
 Containers: 1
  Running: 1
  Paused: 0
  Stopped: 0
 Images: 9
 Server Version: 23.0.5
 Storage Driver: overlay2
  Backing Filesystem: extfs
  Supports d_type: true
  Using metacopy: false
  Native Overlay Diff: true
  userxattr: false
 Logging Driver: json-file
 Cgroup Driver: cgroupfs
 Cgroup Version: 2
 Plugins:
  Volume: local
  Network: bridge host ipvlan macvlan null overlay
  Log: awslogs fluentd gcplogs gelf journald json-file local logentries splunk syslog
 Swarm: inactive
 Runtimes: io.containerd.runc.v2 runc
 Default Runtime: runc
 Init Binary: docker-init
 containerd version: 2806fc1057397dbaeefbea0e4e17bddfbd388f38
 runc version: v1.1.5-0-gf19387a
 init version: de40ad0
 Security Options:
  seccomp
   Profile: builtin
  cgroupns
 Kernel Version: 5.15.49-linuxkit
 Operating System: Docker Desktop
 OSType: linux
 Architecture: aarch64
 CPUs: 5
 Total Memory: 7.667GiB
 Name: docker-desktop
 ID: 33b59bfc-c879-4cc7-b857-4ce95a278567
 Docker Root Dir: /var/lib/docker
 Debug Mode: false
 HTTP Proxy: http.docker.internal:3128
 HTTPS Proxy: http.docker.internal:3128
 No Proxy: hubproxy.docker.internal
 Registry: https://index.docker.io/v1/
 Experimental: false
 Insecure Registries:
  hubproxy.docker.internal:5555
  127.0.0.0/8
 Live Restore Enabled: false

Additional Info

see https://github.com/docker/compose/issues/10310 for context

Start by tracing the Docker Engine implementation of the docker restart command and how State.Restarting is exposed through the inspect API. Reproduce the race with the commands in the issue, then add coverage for the restart-in-progress state; done means inspect reports true while a manually restarted container is restarting.