nuxtServerInit like implementation for Pinia

auth.js

import { defineStore } from 'pinia'

export const useAuthStore = defineStore({
  id: 'auth',
  state: () => ({
    isAuthenticated: false,
    user: null
  }),
  actions: {
      async nuxtServerInit() {
          const user = await this.$axios.get('/api/user')
          this.$patch({
            isAuthenticated: true,
            user
          })
      }
  }
})

nuxt-server-init.js

import { useAuthStore } from '~/store/auth'

export default async function ({ $pinia }) {
    if (process.server) {
        const store = useAuthStore($pinia)
        await store.nuxtServerInit()
    }
}

// Add this file to nuxt.config.js
// { router: { middleware: ['nuxt-server-init'] } }

You should use nuxt plugins instead of middleware.
Because this middware will be invoked every time route changes (of course you already check process.server, but a plugin should be more suitable)

A plugin won't wait for the data to be fetch before loading the page?

I don't know if it's just the recent versions of nuxt but I'm not being able to call useStore inside defineNuxtPlugin, neither access a valid $pinia on nuxtApp param

@LeonardoRick yes you can.
in the nuxt.config we can import store so that it's available throughout the application
imports: { dirs: ['./stores'] }

then the plugins/......

export default defineNuxtPlugins((nuxtApp) => {
   const exampleStore = useExampleStore()
})