Make panna cotta: Pour lemon juice into a bowl. Sprinkle gelatin over and let stand until gelatin softens, 5 to 10 minutes.
In a saucepan, whisk sugar with lemon zest. Pour in half-and-half and yogurt, place over medium heat and cook, whisking, until mixture is just simmering and sugar has dissolved, about 2 minutes (do not allow to boil). Add gelatin mixture and vanilla and whisk until gelatin has dissolved. Transfer to a large measuring cup with a pouring spout and divide mixture among 6 wineglasses, ramekins or custard cups. Refrigerate until firm, at least 4 hours or overnight.
Make compote: Stir together berries, sugar, honey and lemon juice in a saucepan. Cook over medium heat, stirring often, until sugar dissolves and berries release their juice, about 3 minutes. Lower heat slightly and continue cooking, stirring often, until compote thickens slightly, about 5 minutes longer. Transfer to a bowl and cool, then cover and refrigerate. Spoon over panna cotta just before serving.