Combine three-quarters of the butter (3 sticks) with the oil in a heavy bottomed saucepan or Dutch oven. Add the potatoes, and over low heat, bring the butter/oil mixture to temperature at 140°F (use a candy thermometer). Slow-poach the potatoes until tender when pierced with the tip of a paring knife. When ready, remove them from the fat mixture with a slotted spoon, and place on a baking sheet. Salt generously and let cool.
Meanwhile, blanch the leeks in salted, boiling water. Drain and chop. Slice the cooled potatoes into 1/4-inch-thick coins.
In a large saute pan, add just enough canola oil to coat the bottom. Heat over medium-high heat. When the oil is hot, add the sliced potatoes, and saute until beginning to brown. Add the leeks, remaining butter, minced shallot, and parsley and continue to cook until the potatoes are golden brown and slightly crisp on the outside. Meanwhile, heat a grill over high heat for the ramps.
When the grill is ready, toss the ramps in a bowl with a small amount of canola oil, salt, and pepper. Grill the ramps whole until tender.
To serve, place a portion of warm potatoes on each plate and top with slices of Taleggio, so the cheese melts. Top with a grilled ramp and serve immediately.