Attach deep fryer/candy thermometer to side of medium saucepan; add oil and heat over medium heat to 380°.
Prepare Donut Fries: After cutting out donuts, reroll remaining dough into 8 x 12-inch rectangle. With pizza cutter, cut dough lengthwise in half, then cut crosswise into 1/3-inch strips; let stand 15 minutes.
Prepare White Chocolate-Matcha Dipping Sauce: In small saucepot, heat half and half to a simmer over medium heat; remove from heat and whisk in chocolate chips and butter. Add matcha powder; whisk vigorously or blend with an immersion blender until smooth. Makes about 1-1/4 cups.
In batches, gently add donut fries; cook 1 minute or until golden brown, turning once; transfer to paper towel-lined cookie sheets to drain. Serve with warmed White Chocolate-Matcha Sauce caramel sauce or hazelnut spread for dipping.