Preheat the oven to 350°F and place a rack in the middle position. Set a large pot or kettle of water on the stove to come to a boil for the water bath (you will pour this into a larger pan later).
Ensure the cream cheese and eggs are at room temperature.
In the bowl of a stand mixer fitted with the paddle attachment (or using a hand mixer), combine the granulated sugar and lemon zest. Mix until the sugar is evenly moistened and fragrant.
Add the cream cheese and beat until the mixture is smooth and no large lumps remain, stopping to scrape down the bowl as needed.
Add the eggs one at a time, fully incorporating each egg before adding the next. Scrape down the bowl between eggs to keep the batter even.
Add the heavy cream and vanilla extract and mix just until the batter is smooth and homogenous. Scrape the bowl and paddle once more to ensure everything is combined.
Place clean canning jars (lids and bands removed) on a baking sheet or tray and pour the cheesecake batter into the jars until each is about 3/4 full.
Set the filled jars in a larger baking pan or roasting pan. Carefully pour the boiling or very-hot water from the pot into the larger pan until the water reaches about halfway up the sides of the jars (pour slowly to avoid splashing into the jars).
Bake for 25 to 30 minutes, until the edges of each cheesecake appear set but the centers still have a slight jiggle.
Carefully remove the larger pan from the oven. Using oven mitts or a towel, lift the jars out of the water bath and place them on a cooling rack. Allow the jars to cool completely to room temperature.
Once cooled, refrigerate the jars upright (with lids on) for at least 5 hours to fully set.
Before serving, top each cheesecake with the fresh berries and serve.