Place 2 tsp. garlic and 1 Tbsp. ginger in a medium bowl. Add nut butter, sugar, fish sauce, if desired, cayenne, cumin, 2 Tbsp. lemon juice and 1/4 cup water; stir to combine. Set sauce aside.
In a large nonstick skillet, warm 1 Tbsp. oil over medium-high heat. Season chicken all over with salt and pepper. Add half of chicken to skillet and cook until seared on one side, about 2 minutes. Turn and cook until cooked through, about 1 minute longer. Transfer to a plate. Repeat with another 1 Tbsp. oil and remaining chicken. Reserve pan and oil.
Add remaining 1 Tbsp. oil to skillet, then add broccolini, onion and 2 Tbsp. water. Cover skillet and steam until water evaporates and broccolini is crisp-tender, 3 to 4 minutes. Uncover, add remaining garlic and ginger and sauté until they are fragrant and turn lightly golden, 1 to 2 minutes. Stir in almonds, remaining 1 Tbsp. lemon juice and chicken. Divide stir-fry mixture among 4 plates and drizzle with sauce. Serve with rice, if desired.