Preheat oven to 350°F. Line 12 muffin cup holes with liners.
Stir together graham cracker crumbs, brown sugar, and melted butter. Press about 2-2 1/2 tablespoons of the mixture into the bottom of each muffin liner. (I find that using a small shot glass or similar object can help pack the crusts easily.)
Beat room temperature cream cheese with a hand mixer until fluffy. Beat in room temperature sour cream. Beat in egg, sugar, and vanilla bean paste until (mostly) no lumps remain. You can use pure vanilla extract, but using the paste adds lots of flavor and the little vanilla bean flecks. (Room temperature ingredients are a must to avoid lumps, believe me!)
Evenly divide (about 2-2 1/2 tablespoons each) cheesecake among crusts. Bake for 18-20 minutes, until they just begin to brown. Cool completely and refrigerate for at least 3 hours or overnight.
To make the topping, place the lime juice, sugar, and cornstarch in a small saucepan over medium-low heat. Stir until it boils, then cook for 1 minute. Remove from heat and let cool to room temperature. It will thicken as it cools.
Assemble the cheesecakes close to serving time. Brush the top of each cheesecake with a bit of the lime syrup. Place berries on top and brush the berries with more syrup. Chill until ready to serve. (The fruit is best added the same day as serving.) The fruit won’t stick to the cheesecake - it’s more of just a topping.
Store loosely covered in refrigerator. Cheesecakes will last up to 4 days in refrigerator or can be frozen for up to one month. Once you add the fruit, eat within the day.