To make the crusts, 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 pull 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 strawberries, orange liqueur, and sugar in a medium bowl, and toss to coat evenly.
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, on a pizza paddle or large chopping board.
Spread half of the goat cheese mixture across each piece of crust using a rubber spatula. Spread half of the berries across the top, leaving the liquid in the bottom of the bowl.
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. Drizzle each pizza with 1 teaspoon of the remaining honey and serve.
Alternately, place a pizza stone in the oven and preheat to 500 degrees F.
Remove the dough from the refrigerator and roll 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 berries across the top of the pizza, leaving the liquid in the bottom of the bowl. Repeat with the remaining dough, goat cheese, and strawberries.
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, and drizzle each with 1 teaspoon of the remaining honey. Serve.
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