workflows/docker: use GitHub Packages for caching.

The GitHub Actions caching seems to not be very performant and is
exploding our cache storage usage.

.github/workflows/docker.yml

@@ -12,6 +12,7 @@ on:
 
 permissions:
   contents: read
+  packages: write
 
 defaults:
   run:
@@ -102,21 +103,27 @@ jobs:
             fi
           } | tee -a "${GITHUB_OUTPUT}"
 
+      - name: Log in to GitHub Packages
+        uses: docker/login-action@9780b0c442fbb1117ed29e0efdff1e18412f7567 # v3
+        with:
+          registry: ghcr.io
+          username: github-actions[bot]
+          password: ${{ secrets.GITHUB_TOKEN }}
+
       - name: Build Docker image
         uses: docker/build-push-action@4f58ea79222b3b9dc2c8bbdd6debcef730109a75 # v6
         with:
           context: .
           load: true
           tags: brew
-          cache-from: type=gha
-          cache-to: type=gha,mode=max
+          cache-from: type=registry,ref=ghcr.io/homebrew/ubuntu${{ matrix.version }}:cache
           build-args: version=${{ matrix.version }}
           labels: ${{ steps.attributes.outputs.labels }}
 
       - name: Run brew test-bot --only-setup
         run: docker run --rm brew brew test-bot --only-setup
 
-      - name: Log in to GitHub Packages
+      - name: Log in to GitHub Packages (again)
         if: steps.attributes.outputs.push == 'true'
         uses: docker/login-action@9780b0c442fbb1117ed29e0efdff1e18412f7567 # v3
         with:
@@ -138,7 +145,7 @@ jobs:
           context: .
           push: true
           tags: ${{ steps.attributes.outputs.tags }}
-          cache-from: type=gha
-          cache-to: type=gha,mode=max
+          cache-from: type=registry,ref=ghcr.io/homebrew/ubuntu${{ matrix.version }}:cache
+          cache-to: type=registry,ref=ghcr.io/homebrew/ubuntu${{ matrix.version }}:cache,mode=max
           build-args: version=${{ matrix.version }}
           labels: ${{ steps.attributes.outputs.labels }}