Using a vegetable peeler, remove the zest from the limes and place in the bowl of a food processor. Add 1½ cups of sugar and pulse until the zest is finely chopped.
Beat the butter and lime-sugar mixture in the bowl of an electric mixer on medium speed for 2 to 3 minutes, until light and fluffy. Add the eggs, one at a time, blending well after each addition. Add the lime juice and salt and blend until smooth.
Transfer the mixture to a medium saucepan and place over low heat. Cook for about 10 minutes, stirring constantly, until a candy thermometer registers 180°F. Remove from the heat and transfer to a clean bowl. Cover with plastic wrap, pressing the wrap directly onto the surface of the lime curd to prevent a skin from forming. Chill in the refrigerator for at least 2 hours, until thickened.
Combine the cranberries, water, and remaining 2 tablespoons of sugar in a small saucepan over low heat and cook for about 20 minutes, until the cranberries burst and the mixture thickens. Remove from the heat and let cool to room temperature.
Whisk together the flours and salt in a medium bowl. Beat the butter and powdered sugar in the bowl of an electric mixer on medium speed for 2 to 3 minutes, until well combined. Add the egg and egg yolk, one at a time, blending well after each addition. With the mixer on low speed, add the flour mixture in two additions, blending until just incorporated.
Transfer the dough to a lightly floured work surface. Divide the dough in half and shape each into a flat disk. Wrap in plastic wrap and chill in the refrigerator for at least 1 hour, or overnight.
To assemble, spread a thin layer of cranberry compote over the bottom of each tartlet. Spoon the lime curd into the shells. Chill in the refrigerator for 30 minutes before serving.