Make poaching liquid by combining first 5 ingredients in a medium pot. Peel one of the pears. Cut in half lengthwise and carefully scoop out the core. Set pear halves in poaching liquid. Repeat these steps with second pear.
Bring poaching liquid to a simmer, and simmer pears until just tender, about 8 to 10 minutes. Remove from the heat and let pears cool to room temperature in the liquid. Cover and refrigerate pears in the liquid until needed (pears can be made a day in advance of serving).
To make sauce, place raspberries in a food processor, or in the cup that came your immersion (hand) blender. Add 1/4 cup of the pear poaching liquid. Process raspberries until well blended. Strain the mixture through a fine sieve into a small bowl, using the back of a small ladle to squeeze out as much liquid as you can.
To serve, place some sauce on the bottom of 4 dessert plates. Set a pear half, cut-side-up, on each plate. Garnish each pear with a dollop of yogurt and a mint spring, and then serve.