You could define the env variable as follow (remember to put string literal in single quotes):

env:
  GODOT_MONO_BUILD_TAG: ${{ github.event.inputs.mono-build-repo || 'latest' }}

where latest should be the default value for the env var

name: Project Specific Workflow
on:
  push:
    branches: []
  pull_request:
    branches: []
  workflow_dispatch:
    # Defaults here are for worflow_dispatch only, all other events are set in EnvSetup.setup-output-defaults:
    inputs:
      build-and-verify:
        description: 'Do a build and verify'
        required: true
        default: 'true'    
jobs:
  EnvSetup:
    name: Setup Dynamic Environment Variables
    runs-on: 
    outputs:
      build-and-verify: ${{ steps.set-output-defaults.outputs.build-and-verify }}
    steps:
      - name: set outputs with default values
        id: set-output-defaults
        run: |    
          # If workflow_dispatch, use inputs (left), if other trigger, use default env (right)
          echo "::set-output name=build-and-verify::${{ github.event.inputs.build-and-verify || 'true' }}"
  invoke-workflow:
    needs: [EnvSetup]
    uses: //.github/workflows/@
    with:
      build-and-verify: ${{needs.EnvSetup.outputs.build-and-verify}}
    secrets:
      token: "${{ secrets.PERSONAL_ACCESS_TOKEN }}"

Some important things to note about this code snippet that calls a reusable workflow:

1. The “uses” statement that calls the reusable workflow can’t evaluate expressions yet. That means the repo name, branch, etc. must be hardcoded.
2. The “with” statement that supplies arguments can only take elements from the github context and the “needs” object, so you MUST have a line that “needs” the job where you set the values and return the outputs.
3. the echo "::set-output name=::${{}} syntax is where the variable is set and exposed as a step output, then it is returned with the job output that references it. Pay attention to the names of each object to see how they fit together.
4. Because you can’t pass ENV variables to the reusable workflow, they are almost useless in this pattern.
5. I’m hoping GitHub comes up with a better way to assign values inline or pass them into reusable workflows, because handling each parameter three or more times just to pass it into a reusable workflow is cumbersome.
6. I included some extra trimmings to show how to pass secrets to the reusable workflow and evaluate the default vs. an input inline during variable assignment. The problem is that your default from the “inputs” section of “workflow_dispatch” only gets assigned on workflow_dispatch, and not on any other trigger. So the ${{ $var || value }} syntax lets you assign the value on the right if the one on the left is empty.

Asked and answered: actions/runner#455

Final Answer: it was a deliberate choice to NOT populate the ${{ env.* }} space with a runner’s environment.