thing.rb

class Thing
  def execute
    stubbed_method
  rescue MyException
    retry  
  end
  
  def stubbed_method
    # do stuff
  end
end

thing_spec.rb

describe Thing
  it 'does something' do
    thing.stub(:stubbed_method, &on_first_invocation_raise(MyException)

    thing.execute

    expect(thing).to have_received(:stubbed_method).twice
  end

  private

  def on_first_invocation_raise(exception = StandardError)
    lambda do
      unless @invoked
        @invoked = true
        raise exception
      end
    end
  end
end