Cut potato into 1/2"-thick fries. Place fries on prepared baking sheet in a single layer and spray with cooking spray. Tightly cover with a second piece of foil. Bake 20 minutes. Fries will finish cooking in a later step. While fries bake, mince garlic. Stem and mince oregano. Halve mushrooms. (Quarter if larger than a ping-pong ball.)

Place a medium non-stick pan over medium-high heat and add 1 tsp. olive oil. Add half the garlic (reserve remaining for fries) to hot pan and cook until golden and aromatic, 30-90 seconds. Stir in crushed tomatoes, half the oregano (reserve remaining for fries), sugar, 1/4 tsp. salt, and a pinch of pepper. Add red pepper flakes (to taste). Cook until warmed through, 1-2 minutes. Remove from burner.

Line a plate with a paper towel. Place another medium non-stick pan over medium-high heat and add canola oil. Combine tempura mix and 1/2 cup cold water in a mixing bowl. Add cold water, 1 Tbsp. at a time, until batter is very thin. Coat mushrooms in batter completely.

Test oil temperature by adding a drip of batter to it. It should sizzle gently. If it browns immediately, turn heat down and let oil cool. Working in batches, carefully place mushrooms in hot oil and fry, turning occasionally, until golden brown and crispy, 4-6 minutes. Mushrooms will not be fully submerged in oil. Remove to towel-lined plate.

Immediately return pan used to fry mushrooms, with oil, to medium-high heat. Add fries to hot oil and flip occasionally until golden brown and crispy, 4-6 minutes. Transfer to another mixing bowl with remaining oregano, remaining garlic (to taste), and 1/4 tsp. salt and toss to coat. While potatoes fry, warm rolls directly on oven rack, 3-5 minutes.

Spoon 2 Tbsp. sauce on inside of each roll. Arrange fried mushrooms on top of sauce. Place sandwich on a plate and serve fries and remaining sauce alongside.