Preheat the oven to 375 degrees F.
Slice the pears lengthwise into halves. With a small sharp paring knife and a melon baller, remove the core and seeds from each pear, leaving a round well for the filling and add some butter.
Trim a small slice away from the rounded sides of each pear half so that they will sit in the baking dish without wobbling.
Toss the pears with some lemon juice to prevent them from turning brown. Arrange them, core side up, in a baking dish large enough to hold the pears snugly.
Gently toss the crumbled blue cheese, dried cranberries, and walnuts together in a small bowl. Divide the mixture among the pears, mounding it on top of the indentation.
In the same small bowl, combine the white wine, dark rum and honey, stirring to dissolve the honey. Pour the mixture over and around the pears.
Bake the pears, basting occasionally with the wine mixture, for 30 minutes, or until tender. Set aside until warm or at room temperature.
If you like, just before serving, whisk together the olive oil, lemon juice, and 1/4 cup of the basting liquid in a bowl. Divide the arugula among 6 plates and top each with a pear half. Drizzle each pear with some of the basting liquid, sprinkle with salt, arrange the caramelized walnut halves and serve warm.