Peel and core the pears. Combine 2½ cups of the sugar and 5 cups water in a saucepan large enough to hold the pears. Split the vanilla beans the long way and scrape out the seeds; add both seeds and pods to the water. Turn the heat to medium-high and bring to a boil.

Add the pears and adjust the heat so that the mixture bubbles, but not too vigorously. Cook for 8 minutes, or until a thin-bladed knife inserted into the pears meets with little resistance. Let the pears cool in the liquid for 30 to 60 minutes (do not refrigerate).

Meanwhile, combine 1 cup water with the remaining ¾ cup sugar in a small saucepan; bring to a boil, stirring to dissolve the sugar. Turn off the heat and whisk in the cocoa, along with the cream. Return to very low heat and cook, stirring, just until thickened slightly. (This can be made several hours in advance; keep at room temperature, then rewarm gently.)

Serve the pears with the warm chocolate sauce spooned over them.

The pears should be large and just about perfectly ripe before cooking. To judge ripeness, gently squeeze their "shoulders," which should yield to your touch.

The easiest way to core a pear is with a small melon baller, digging up from the bottom. An ordinary teaspoon works almost as well.