[ci] Also cache playground playwright browsers

Following #32678, do the same for the playground e2e test since this step can sometimes take many minutes to complete.

Author: poteto

.github/workflows/compiler_playground.yml

@@ -39,8 +39,21 @@ jobs:
           path: |
             **/node_modules
           key: compiler-node_modules-v6-${{ runner.arch }}-${{ runner.os }}-${{ hashFiles('compiler/**/yarn.lock') }}
+      - name: Store Playwright version
+        id: playwright_version
+        run: |
+          PLAYWRIGHT_VERSION=$(npm ls @playwright/test | grep @playwright | sed 's/.*@//')
+          echo "Playwright's Version: $PLAYWRIGHT_VERSION"
+          echo "playwright_version=$PLAYWRIGHT_VERSION" >> "$GITHUB_OUTPUT"
+      - name: Cache Playwright Browsers for Playwright's Version
+        id: cache_playwright_browsers
+        uses: actions/cache@v4
+        with:
+          path: ~/.cache/ms-playwright
+          key: playwright-browsers-v6-${{ runner.arch }}-${{ runner.os }}-${{ steps.playwright_version.outputs.playwright_version }}
       - run: yarn install --frozen-lockfile
       - run: npx playwright install --with-deps chromium
+        if: steps.cache_playwright_browsers.outputs.cache-hit != 'true'
       - run: CI=true yarn test
       - run: ls -R test-results
         if: '!cancelled()'