thing_spec.rb

class Thing
  def execute
    stubbed_method
  end
end

MyException = Class.new(StandardError)

describe Thing do
  subject(:thing) { Thing.new }

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

    expect { thing.execute }.to raise_error(MyException)

    expect { thing.execute }.to_not raise_error

    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