For the dip: In a food processor, add the cream cheese, dill, cucumber and 1 teaspoon of the lemon juice and pulse until smooth. Taste and add more lemon juice if needed. Season with salt and pepper. Refrigerate until ready to serve.
For the eggplant: Place the eggplant spears on a baking sheet and lightly brush with the oil just to coat. In a large bowl, whisk together the flour, Greek seasoning, a pinch of salt and a few grinds of pepper. Add the eggplant in batches and coat with the flour mixture, shaking to remove the excess, then remove to a plate.
For the batter: In a large bowl, whisk together the flour, Greek seasoning, lemon zest and a nice pinch of salt until combined. Slowly whisk in the beer until you achieve a pancake batter consistency; you may not use all of the beer. Allow the batter to rest 5 minutes. If it thickens, loosen with a bit more beer.
Heat a fryer filled with oil to the proper level or a pot with about 2 inches of oil to 365 degrees F.
In batches, dunk the flour-coated eggplant spears into the batter, shake off the excess and gently place in the oil. Fry until golden brown, 4 to 6 minutes, making sure to maintain an oil temperature around 350 degrees F. Remove the eggplant to a paper towel-lined plate and immediately spritz lightly with lemon juice and sprinkle with a pinch of salt. Serve immediately with the chilled dip.