Ramban offers a kabbalistic reading, explaining that the Urim and Tumim functioned through the divine light (or) that would shine through the High Priest's heart, transforming him into a vessel of divine illumination. He disagrees with Rambam's naturalistic account and insists the process was entirely miraculous and dependent on the holiness of the garments and the wearer.
וְהָעִנְיָן הוּא, כִּי הָיוּ שֵׁמוֹת קְדוֹשִׁים, מִכֹּחָם יָאִירוּ הָאוֹתִיּוֹת מֵאַבְנֵי הַחֹשֶׁן אֶל עֵינֵי הַכֹּהֵן הַשּׁוֹאֵל בְּמִשְׁפָּטָם (במדבר כז כא) וְהַמָּשָׁל, כִּי כַּאֲשֶׁר שָׁאֲלוּ "מִי יַעֲלֶה לָּנוּ אֶל הַכְּנַעֲנִי בַּתְּחִלָּה לְהִלָּחֶם" (שופטים א א). הָיָה הַכֹּהֵן מְכַוֵּן בַּשֵּׁמוֹת שֶׁהֵם הָאוּרִים, וְהֵאִירוּ לְעֵינָיו אוֹתִיּוֹת יְהוּדָה, וְיוֹ"ד מִלֵּוִי, וְעַיִ"ן מִשִּׁמְעוֹן, וְלָמֶ"ד מִלֵּוִי, וְהֵ"א מֵאַבְרָהָם הַכָּתוּב שָׁם עַל דַּעַת רַבּוֹתֵינוּ (יומא עג:). וְהִנֵּה כַּאֲשֶׁר הָאוֹתִיּוֹת מְאִירוֹת אֶל עֵינֵי הַכֹּהֵן עֲדַיִן לֹא יָדַע סִדּוּרָן, כִּי מִן הָאוֹתִיּוֹת אֲשֶׁר סִדְּרוּ מֵהֶן "יְהוּדָה יַעֲלֶה" הָיָה אֶפְשָׁר לְהֵעָשׂוֹת מֵהֶם "הויהד עליה", אוֹ "היעל ידוהה", וְתֵבוֹת אֲחֵרוֹת רַבּוֹת מְאֹד, אֲבָל הָיוּ שָׁם שְׁמוֹת הַקֹּדֶשׁ אֲחֵרִים נִקְרָאִים "תֻּמִּים" מִכֹּחָם יִהְיֶה לֵב הַכֹּהֵן תָּמִים בִּידִיעַת עִנְיַן הָאוֹתִיּוֹת שֶׁהֵאִירוּ לְעֵינָיו.
They were thus of heavenly origin, and therefore they are referred to without any specification and with the definite article, in a similar usage to that which we have in the verse, and He placed at the east of the garden of Eden the cherubim. Now Moses took the inscription of the Urim and the Thummim, and placed them in the breastplate of judgment after he had clothed Aaron with the ephod and the breastplate, as it is said, And he put the ephod upon him, and he girded him with the skilfully-woven band of the ephod… And he placed the breastplate upon him; and in the breastplate he put the Urim and the Thummim. For only after [Aaron was already clothed with the ephod and the breastplate] did Moses place the Urim and the Thummim between the folds of the breastplate. Thus the Urim and the Thummim were the holy Names of G-d, and it was by virtue of the power residing in these Names that the letters inscribed upon the stones of the breastplate would light up before the eyes of the priest who inquired of their judgment. For example, when they inquired, Who shall go up for us first against the Canaanites, to fight against them, the priest fixed his thoughts on those Divine Names which were the Urim [literally: “lights”], and the letters forming the name Yehudah lighted up before his eyes, and [for the word ya’aleh — “he shall go up”] the letter yod lighted up from the word Levi, the ayin from Shimon the lamed from Levi, and the hei from Avraham which was also written there, according to the opinion of our Rabbis, or perhaps the hei from Yehudah lighted up a second time. Now when the letters lighted up before the eyes of the priest, he did not yet know their arrangement [that is, how these letters were to be grouped together into words], for from the letters forming the words Yehudah ya’aleh (Judah shall go up), it is possible to form the words: hoy heid alehah, or hie al Yehudah, and very many other words.