Warm the canola oil and butter in large saute pan over medium heat. Add the mushrooms and cook slowly, stirring frequently, for 5 minutes, or until the mushrooms release their juices. Continue cooking until all their moisture evaporates.
Add the shallots and cook for 3 minutes. Strip the leaves from the thyme sprigs and add to the pan; discard the stems. Season with the salt and pepper. Remove from heat.
Place the potatoes in a large bowl. Drizzle with the olive oil and toss until well coated.
Place 2 medium nonstick saute pans over medium heat. Place 1 potato slice in the center of each pan. Use one-quarter of the potatoes per pan, carefully form circles of slightly overlapping potato slices around the center slices, letting them fall slightly over the edge of the pan. Season with salt and pepper. Cook for 4 minutes. If any gaps form between the slices, patch with slices cut to fit the gap.
Place 1/4 of the mushroom mixture in the center of each galette and evenly spread it out to cover the potatoes. As the potatoes become soft and flexible, begin folding them inward so that they slightly enclose the mushrooms and form a perfect disk-shaped galette. Cook for 25 minutes, turning occasionally so that the potatoes do not burn or become to brown.
Serve once the galette is crisp and holds its shape, warm or at room temperature.