To make the crust, in a medium bowl, combine the yeast, water, and 1 tablespoon of the flour. Whisk to blend. Let sit until the mixture becomes slightly foamy, about 5 minutes.
Combine the remaining flour and salt in a large bowl and stir to blend. Add the yeast mixture, 2 tablespoons of the oil, and the cream. Stir well with a heavy wooden spoon until it begins to come together and pulls away from the sides of the bowl. Turn out onto a lightly floured surface and knead for 4 to 5 minutes into a smooth, but slightly sticky dough. Oil a clean bowl with 1 teaspoon of the oil and turn the dough in the oil to lightly coat. Cover the bowl with plastic wrap or a clean kitchen towel and allow to rise until doubled in size in a warm, draft-free place, about 40 minutes.
Lightly grease a baking sheet with the remaining 1/2 teaspoon of oil. Divide the dough into 2 equal pieces and form into balls. Place on the greased sheet, cover with plastic wrap, and refrigerate until doubled in volume, about 2 hours.
Combine the plums, 2 tablespoons honey, and Armagnac in a bowl and let sit at room temperature for 30 minutes.
Clean a grill very well with a brush, and lightly grease with vegetable oil. Preheat the grill to medium heat.
Combine the goat cheese and 3 tablespoons honey in a bowl and mash with a fork until smooth.
Remove the dough from the refrigerator and roll each ball into a 10-inch round on a lightly floured pizza paddle or large chopping board. Transfer to the grill and cook on 1 side until golden brown, turning with a spatula to mark evenly, 3 to 4 minutes. Remove from the grill and place, cooked side up on a pizza paddle or large chopping board.
Spread half of the goat cheese mixture across each piece of crust using a rubber spatula. Arrange half of the plums across the top of each crust.
Transfer the pizzas to the grill, cover, and cook until golden brown, rotating with a spatula to keep from burning, 3 to 4 minutes. Transfer to serving plates, sprinkle each with a pinch of black pepper and half of the mint leaves. Serve immediately.
Alternately, place a pizza stone in the oven and preheat to 500 degrees F.
Remove the dough from the refrigerator and roll each out into a 10-inch round on a lightly floured pizza paddle or large chopping board. Make "dimples" with your fingertips on the top of the dough. Spread half of the goat cheese mixture across the pizza using a rubber spatula. Spread half of the plums across the top of the pizza. Repeat with the remaining dough, goat cheese, and plums.
Transfer to the pizza stone (or 2 heavy baking sheets) and bake until golden brown, about 12 minutes. Remove from the oven and transfer to serving plates. Sprinkle a pinch of the black pepper and half of the mint leaves over the top of each pizza. Serve immediately.