First off, and not to digress, but it's frustrating when the first ingredient is listed as "1/2 lemon." What is that? 1 ounce? 1/2 ounce? 3/4? Depends on how fresh the lemon is? Let's ignore that for a moment. The 4 dashes of grenadine is 4/6 of a teaspoon. Then, obviously, 2 ounces of applejack. With these proportions, maybe we should say about 1/2 ounce of lemon juice, depending on how tart you might want it. here's a recipe from "Famous New Orleans Drinks (and how to mix 'em)":
Again with the catch-as-catch-can measurement of the lemon juice. They call for about twice the amount of grenadine, but less of the applejack (1.5 ounces). Plus some Peychaud bitters, since this coming from New Orleans, after all. The proportions of spirit to citrus to grenadine can be toyed with, to suit your taste. Without taste-testing it first, maybe we'll recommend something like this: 2 oz of applejack, 1/2 oz lemon juice, and a couple of teaspoons of grenadine (because we like it sweet). Try it for yourself and see what you think.
Sometimes these old and more obscure drinks can trigger some interesting stories. We don't have any of our own to share, but we can share one we read in the New York Times not long ago. It so happens that they did a feature on calvados, and the next week someone wrote a letter to the editor about the article. It turns out that her father had served in France during WWII and came home with a taste for calvados. All his life, the Jack Rose was his drink of choice -- but it had to be made with calvados, not applejack. At every family gathering, Dad was mixing and pouring his calvados Jack Roses. Unfortunately for the daughter, it was a stiffer drink than she liked, and she had to find a way to dispose of it in a potted plant or the bathroom sink. What a shame. But, it was part of loving her Dad. The article of about calvados must have brought it all back to her -- enough for her to take the time to write a letter to the editor to share the memory. Now, we share it with you.