In the bowl of a stand mixer, whisk together flour, thyme, yeast, and salt until combined. Add ¾ cup (180 grams) warm water, oil, and 2 teaspoons (14 grams) honey. Using the paddle attachment, beat at low speed until a shaggy dough forms, 1 to 2 minutes.
Switch to the dough hook attachment. Beat at medium-low speed until dough is smooth and elastic, 10 to 13 minutes. Dough will be slightly tacky but should come away from sides of bowl.
Lightly spray a medium bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let stand in a warm, draft-free place (75°F/24°C) until doubled in size, 45 minutes to 1 hour.
Position oven rack in bottom third of oven. Preheat oven to 450°F (230°C). Place a rimmed baking sheet upside down in oven to preheat for 30 minutes.
In a medium bowl, stir together ricotta, red pepper, and remaining 2 teaspoons (14 grams) honey.
Gently punch dough down, and divide in half. Working with one portion of dough at a time, press and flatten into a 12×9-inch oval or rectangle; place dough on a parchment paper-lined baking sheet. (If dough does not stretch easily, cover and let stand for 10 minutes and then try again.) Top with 6 slices mozzarella. Carefully slide dough on parchment onto preheated baking sheet.
Bake until crust is golden brown and cheese is melted, 10 to 15 minutes. While first flatbread is baking, repeat procedure with remaining dough and remaining cheese.
Top hot flatbreads with peach slices, and dollop with ricotta mixture; top with arugula, thyme, and honey. Serve warm.