Squeeze a lemon half into a large bowl of cold water and add the lemon half to the bowl. Working with 1 artichoke at a time, trim the stem to 1 inch and snap off all of the tough outer leaves. Using a sharp knife, cut off the top two-thirds of the artichoke. With a teaspoon, scoop out the furry choke. Peel the artichoke, removing all the dark green skin and add it to the bowl of acidulated water. Repeat with the remaining artichokes.
Preheat the oven to 425°. Drain the artichokes and pat dry. Cut each artichoke into eighths. Quarter the remaining lemon half and remove the seeds. In the bowl, toss the artichokes with the lemon pieces, garlic, olive oil and thyme. Season with salt and pepper and spread on a large nonstick rimmed baking sheet. Bake the artichokes for about 40 minutes, stirring occasionally, until golden and tender. Let cool slightly and discard the thyme. Coarsely chop 8 of the roasted artichoke pieces and reserve. Reduce the oven temperature to 375°.
Peel the garlic; place in a food processor. Add the remaining roasted artichoke and lemon pieces; pulse until finely chopped. Add the mayonnaise and cream cheese and process until smooth. Season with salt and pepper. Add the reserved chopped artichokes and pulse just until combined. Spread the dip in an even layer in a small shallow baking dish.
In a small bowl, mix the bread crumbs and butter; sprinkle over the dip. Bake for 20 minutes, or until heated through and the topping is golden. Serve with crackers or bagel chips.
The artichoke dip can be prepared through Step 3 and refrigerated for up to 3 days.