In a medium bowl, whisk together the marinade ingredients. Add in the chicken and toss until the pieces are well coated. Let it marinate in the fridge for at least 30 minutes or up to 2 hours.
Preheat your oven to 500°F/260°C and place a pizza stone, a large cast iron skillet, or an inverted rimmed baking sheet on the middle rack.
Heat the olive oil in a large skillet over medium heat. Cook the chicken for 5 minutes on each side or until the internal temperature reaches 165ºF. Remove from the skillet and cut into bite-size pieces.
Make the red curry sauce by whisking the coconut milk and red curry paste together until smooth.
On a lightly floured surface, roll out the pizza dough into a circular shape and then transfer it to a pizza peel or a preheated baking sheet (or cast iron skillet) dusted with flour or cornmeal. If using a gluten-free crust, you may want to put down a piece of parchment paper first for easy removal.
Top each pizza with the red curry sauce. Sprinkle shredded cheese over the sauce followed by the cooked chicken and vegetables.
Bake for 10 minutes. For a golden brown crust, put the cooked pizza under the broiler for an additional 2 minutes, watching closely so it doesn't burn. Remove from the oven and top with cilantro, basil, or sliced green onions.